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In what scenario would you use a function that returns another function (Javascript)?

Thanks to this question I understand how a function may take two sets of parentheses and how a function may return another function which fires immediately after being returned.
What I do not understand is why this is good to know? How could this be applied to solve a problem?
The example of:
function add(x){
return function(y){
return x + y;
};
}
add(3)(4) === 7; // true
Works fine - sure. But why not just write it like this instead?
function add(a, b) {
return a + b;
}
add(3, 4) === 7; // true

Lets take the same code which you have mentioned.
function add(x) {
return function(y) {
return x + y;
};
}
var adder3 = add(3); //Forming adder3
var op1 = adder3(4) // 7
var op1 = adder3(5) // 9
// Now adder 10:
var adder10 = add(10); //Forming adder3
var op1 = adder10(4) // 14
var op1 = adder10(5) // 15;
Hope you understand!!
Revert me if you need more info on closure.

Your example called as closures
Closures’ Rules and Side Effects
Closures have access to the outer function’s variable even after the outer function returns:
One of the most important and ticklish features with closures is that the inner function still has access to the outer function’s variables even after the outer function has returned. Yep, you read that correctly. When functions in JavaScript execute, they use the same scope chain that was in effect when they were created. This means that even after the outer function has returned, the inner function still has access to the outer function’s variables. Therefore, you can call the inner function later in your program. This example demonstrates:
function celebrityName(firstName) {
var nameIntro = "This celebrity is ";
// this inner function has access to the outer function's variables, including the parameter​
function lastName(theLastName) {
return nameIntro + firstName + " " + theLastName;
}
return lastName;
}​​
var mjName = celebrityName("Michael"); // At this juncture, the celebrityName outer function has returned.​
​​ // The closure (lastName) is called here after the outer function has returned above​
​ // Yet, the closure still has access to the outer function's variables and parameter​
mjName("Jackson"); // This celebrity is Michael Jackson

Closures store references to the outer function’s variables; they do not store the actual value. 
Closures get more interesting when the value of the outer function’s variable changes before the closure is called. And this powerful feature can be harnessed in creative ways, such as this private variables example first demonstrated by Douglas Crockford:
function celebrityID() {
var celebrityID = 999;
// We are returning an object with some inner functions​
// All the inner functions have access to the outer function's variables​
return {
getID: function() {
// This inner function will return the UPDATED celebrityID variable​
// It will return the current value of celebrityID, even after the changeTheID function changes it​
return celebrityID;
},
setID: function(theNewID) {
// This inner function will change the outer function's variable anytime​
celebrityID = theNewID;
}
}​
}​​
var mjID = celebrityID(); // At this juncture, the celebrityID outer function has returned.​
mjID.getID(); // 999​
mjID.setID(567); // Changes the outer function's variable​
mjID.getID(); // 567: It returns the updated celebrityId variable


Reference site : http://javascriptissexy.com/understand-javascript-closures-with-ease/

Functions which return functions are useful when you want similar functions which depend on some parameters.
A real life example: [].sort can be called with a custom comparator function, but it can make sense to define a comparator function to allow more customizations:
function comparator(options) { // Function which returns a function
return function(a, b, tmp) {
if(options.reverse) tmp = a, a = b, b = tmp;
if(options.map) a = options.map(a), b = options.map(b);
if(options.func) return options.func(a, b);
return a < b ? -1 : (b < a ? 1 : 0);
}
}
Then you can use
[1,11,10,2].sort(comparator({map: String})); // [1, 10, 11, 2]
[1,11,10,2].sort(comparator({reverse: true})); // [11, 10, 2, 1]
[1,11,10,2].sort(comparator({func: Function.prototype})); // [1, 11, 10, 2]

if we require a function in certain state with certain value then we can use it inside another function and return that, so that the return function with certain state can be directly used in different scenario.
you can check out various example on closure.
http://javascriptissexy.com/understand-javascript-closures-with-ease/

If you know the first param is always going to be the same, then it will convenient to have it closure, instead of passing it again and again. For simple programs, it may not make sense. But, for programs which handles repetitive params more often, this technique definitely comes handy.

Other than closures you can also use it for pre-processing as a one time job, consider if you had to do something intensive, e.g. generate a million things;
function generateSessionSecrets(lock) {
var secrets = [], i = 1000000;
while (i-- > 0) {
secrets[i] = Math.random();
}
return function(key, i) {
if (key === lock) return secrets[i];
}
}
var chest = generateSessionSecrets('fizz');
chest('fizz', 0); // e.g. 0.2096199430525303
chest('fizz', 1); // e.g. 0.30329699837602675
// ...
chest('fizz', 0); // still 0.2096199430525303
(This is an example of concept, not an example of real security)

there isn't a point to using it immediately. you would use it to create a function to attach it to an event or use as a callback for an asynchronous function. an example might be such:
function factory(param){
return function(result) {
if (result==param) dosomething();
}
}
$('#domobject').click({
param = $('#domvalue').value;
asynch_function(factory(param));
});
Here I've attached a click event to presumably a button. When it is clicked it will retrieve the value of an input and create a function based on it and call an asynchronous function with the newly created function as it's callback. The asynchronous function might be an ajax request. When the asynchronous function completes the function that factory created, the callback, will be called. It will check the return value the asynchronous function passed to the callback against the param specified when the event was attached.
If we move the dom lookup to inside the callback function, then we wouldn't need factory or param, but then it would use the value that is in the input at the time when the asynch function has returned, rather than when the button was clicked which is later and the value might have changed.
Sometimes, you won't be able to obtain a value you need in the context of the callback for other reasons. Or it might just be that you want to abstract out a class of functions so you don't have to retype a slightly different version of it in all the places you use it.

Is it possible to change scope of function after new Function()?

So I have this code:
function fn(a){
var f=(new Function("return a"));
return f();
}
fn(7)//ReferenceError: a is not defined
Same problems with local variables:
function fn(){
var a=7;
var f=new Function("return a");
return f();
}
fn(7)//ReferenceError: a is not defined
I want it to return a but the new function cant see a,
it can only see global a
var a=1;
function fn(a){
var f=(new Function("return a"));
return f();
}
fn(7)//1
With the normal initialization the function can see the argument.
function fn(a){
var f=function(){return a};
return f();
}
fn(7)//7
I need to call the basic constructor in my project and can't use global variables.
I know that i could solve this by giving arguments to the newly created function and call it with that like this:
function fn(a){
var f=(new Function('a',"return a"));
return f(a);
}
fn(7)//7
And also could use some parsing function and some stupidly long method to make incoming arguments reachable like this:
function parsargs(funct){
//some parsing methodes giving back argument name list from funct.toString()
return "['a','b']";//like this
}
function fn(a,b){
var arrgstr,retfunc;
arrgstr="";
for(var i in arguments)
{
if(i<arguments.length-1)
arrgstr+=arguments[i]+",";
else
arrgstr+=arguments[i];
}
//return arrgstr;
retfunc="var f=new Function("+parsargs()+",'return b*a');return f("+arrgstr+")";
return (new Function(retfunc))();
}
fn(7,4)//28
But there must be an easier way which reaches local variables and functions as well...
Any suggestions?
PS:
i am trying to replace eval() in the project
Here is a simplified version of my original problem:
fiddle
The answer is NO...

Your exact question isn't clear but, supposing you can use arguments.callee (i.e. non strict mode) and that you want to be able to have any arguments name in fn, you "may" do this:
function fn(a,b){
var strargs = arguments.callee.toString().match(/\(([^\)]*)\)/)[1];
return (new Function(strargs.split(","),"return a+b")).apply(null,arguments);
}
console.log(fn(7, 3)) // 10
But I have a strong feeling this is a XY question and that we could have given a more useful answer knowing the real original problem to solve.

You could call your new Function with a context that references the local variables that you need:
function f(a) {
var b = 30;
return new Function("return this.a + this.b").call({ a: a, b: b })
}
f(10) // 40

The reason why this isn't working as easy as you would like is that JavaScript doesn't blindly put variables in a scope. Instead, it parses the function body (as good as it can) and determines which variables the code will need. It will then look up the variables in the outer scopes, create references for them and attach those references to the new function. That's what keeps variables from the outer scope accessible inside of the function when it eventually will be executed.
You want to build a function from a string body. JavaScript can't tell which variables the body will use and hence, it won't make those variables available.
To make it work for function arguments, use apply()::
function fn(a){
var argref = arguments; // create reference to the arguments of fn() and hence to a or any other arguments
var func = new Function("return arguments[0]");
var wrapper = function() {
return func.apply(null, argref);
};
return wrapper;
}
Note that you still can't reference arguments by name since the code never specifies what the names of the arguments of func are - JavaScript can't magically read you mind and do what you want. If you want to access the arguments by name, you need to tell the interpreter the names.
This question has some code how to determine the names from a function reference: How to get function parameter names/values dynamically from javascript
I don't see a way to make local variables available without passing them to fn() as arguments. Even if you used them inside of fn(), they would be out of scope when func() is eventually executed.
A simple solution would be to pass an object to fn():
function fn(conf) {
var func = new Function('conf', "return conf.a");
var wrapper = function(conf) {
return func.apply(null, conf);
};
return wrapper;
}
fn({a:7});

new Function doesn't create a closure context, but eval does.
Here's a low-tech way to build a function with an arbitrary evaluated body and access to the calling scope:
function f(a) {
return eval("(function() { return a })")()
}
f(10) // 10

Understanding Higher Order functions in Javascript

I am currently reading Eloquent Javascript Chapter 5. They give the following example which is confusing the hell out of me.
function greaterThan(n) {
return function(m) { return m > n; };
}
var greaterThan10 = greaterThan(10);
console.log(greaterThan10(11));
// → true
Can anyone break this down to me as simply as possible. I have huge trouble with callbacks. Especially when it comes to situations like these.

Higher-Order functions basically mean two things:
Functions can take other functions as an argument/input
Functions can return functions
This is what is meant by higher-order functions.
// this function takes a function as an argument
function myFunc(anotherFunc) {
// executes and returns its result as the output which happens to be a function (myFunc)
return anotherFunc();
}
// let's call myFunc with an anonymous function
myFunc(function() {
// this returns a function as you see
return myFunc;
});
As for your example, it demonstrates higher-order functions by returning a function. It also demonstrates the notion of closure.
Closure is closing over a scoped variable, in this case the input argument n.
function greaterThan(n) {
// n is closed over (embedded into and accessible within) the function returned below
return function(m) { return m > n; };
}
// greatherThan10 reference points to the function returned by the greaterThan function
// with n set to 10
// Notice how greaterThan10 can reference the n variable and no-one else can
// this is a closure
var greaterThan10 = greaterThan(10);
console.log(greaterThan10(11));
// → true

There's no "callback" involved here. What you've got with the "greaterThan" function is a function that returns another function.
So, you call the function:
var greaterThan10 = greaterThan(10);
Now the variable "greaterThan10" references the function returned by the "greaterThan" function invoked with 10 as the argument.
Then, you log the result of calling that function:
console.log(greaterThan10(11));
The function returned from "greaterThan" is invoked. It compares its parameter to the value of the parameter "n" passed when it was created. Because 11 is in fact greater than 10, the function will return true and that's what'll be logged.

Javascript, getting away from 'this' and using local scope

In a typical js class, all calls to member functions must be preceded by this. I was looking at a technique that would let me create a library of inter-dependent STATIC functions and relying on closure/scope to make things a bit easier.
Example:
var Singleton={
//main entry point
// call with fn name, args...
call:function(){
var args=[];
if (arguments.length==0) {
return;
}
// get the fn name
var fn=arguments[0];
var x;
// make args array
for (x=1;x<arguments.length;x++) {
args[args.length]=arguments[x];
}
// I want to get rid of this part
// See below for what I wish
// Here I have access to fns below due to hoisting in js
// so I put them in a map...
var fns={
test:test
// etc, more like this I do not want to type/maintain
}
// ... all so I can do this
// get my function.
var fun=fns[fn];
// instead of that, I would like to "override whitespace" and
// say something like:
// var fun=['fn_name'];
// so I can index into local scope and get a fn
//
// log error if not found
if (typeof fun=='undefined') {
loge('Singleton: function not found:'+fn);
return;
}
// ok, run the function
return fun.apply(window,args);
// the test fn accesses test2() without dot notation
function test(a){
// Note: here in test fn it can access test2()
// without using this.test2() syntax
// as you would in normal objects
var s=test2();
alert(s+' test:'+a);
};
function test2(){
return 'test2';
};
}
}
I was hoping someone more familiar with advances in javascript might have advice on how to emulate an "implied but unnecessary this", it always struck me as strange that this defaults to window, and wouldn't it be nice if this could be pointed to an anonymous object with the local scope attached.
I would love to say ['localObject'] to get something in scope.
Edit:
After seeing some of the responses, I will restate this in the form of a challenge:
What I am looking for is a syntax cheat, a way to, as #Varuna put it, "1. Access static methods without using this variable i.e. they will remain global to one another. 2. Do not want to maintain a local array for static methods and want to achieve with in the local scope itself."
Put differently, I need to have the declared functions Register themselves, but I don't want to state the function name more than once. I suppose #Varuna has a solution using eval to access the local scope.
The following approach wont work:
var o={};
o['fn']=function fn(){};
o['fn2']=function fn2(){};
...because you have to state the fn name twice, but closure is preserved.
And this:
var a=[
function fn(){}
,function fn2(){}
];
Register(a);
Will not work because, AFAIK, you lose closure, ie. fn2 cannot see fn. Which also makes the following declarative style a "this nightmare":
window.MINE={
fn:function fn(){
//this?
// want to say fn2(), not this.fn2(), nor MINE.fn2()
}
,fn2:function fn2(){
//this?
}
,deeper:{
//more
}
};
But something like this might work, if you created a weird property that does the registration on assignment:
var registar=new Registar();
registar.reg=function fn(){};
registar.reg=function fn2(){};
//then access
var fn=registar.getFn(n);
// or
var fn=registar._[n];
The above relies on js properties and having access to fn.name, which is not available in all cases AFAIK.

If I understand correctly, you want to create objects that:
have static members
... which can be accessed without using the this notation
The easiest solution (assuming I've properly understood your query), would be to simply use a closure to store your stratic fields, access them directly by name, then explicitly add them as object members.
Consider:
var myConstructor = (function(){
var foo = 'someStaticField';
var bar = function(){
alert('A static method returns ' + foo);
};
return function(){
return {
foo : foo,
bar : bar
};
};
})();
var myInstance = new myConstructor();

As per my understanding, you want to:
1. Access static methods without using this variable i.e. they will remain global to one another.
2. Do not want to maintain a local array for static methods and want to achieve with in the local scope itself.
You can check whether a method exist using eval.Check Here
Only drawback is that this will be using eval method.
Code will be:
var Singleton = {
//main entry point
// call with fn name, args...
call: function () {
var args = [];
if (arguments.length == 0) {
return;
}
// get the fn name
var fn = arguments[0];
var x;
// make args array
for (x = 1; x < arguments.length; x++) {
args[args.length] = arguments[x];
}
//check whether function exist in local scope and not in global scope
if (typeof eval(fn) !== 'undefined' && typeof window[fn] === 'undefined') {
// ok, run the function
return eval(fn).apply(window, args);
}
else{
// log error if not found
loge('Singleton: function not found:' + fn);
return;
}
// the test fn accesses test2() without dot notation
function test(a) {
// Note: here in test fn it can access test2()
// without using this.test2() syntax
// as you would in normal objects
var s = test2();
alert(s + ' test:' + a);
};
function test2() {
return 'test2';
};
}
}

How about declaring functions that can access each other in separate closure, and exporting them to main method by binding your call method to an object containing the functions? Something like previous post (modified slightly):
var Singleton = {
call: (function() {
// here 'call' is bound to object containig your test functions
// this: {test, test2}
if (0 == arguments.length) return;
// log error if not found
if ('function' != typeof this[arguments[0]]) {
console.warn('Singleton: function not found:' + arguments[0]);
return;
}
// '...index into local scope and get function
// ie. get the function by it's name
return this[arguments[0]].
apply(window, Array.prototype.slice.call(arguments, 1));
// --- or:
// you can explicitly introduce function names to current scope,
// by `eval`-ing them here (not very much preferred way in JavaScript world):
for (var fname in this)
if (this.hasOwnProperty(fname))
eval('var ' + fname + ' = ' + this[fname]);
// and you can reference them directly by using their names
var fn = eval(arguments[0]);
return fn.apply(window, Array.prototype.slice.call(arguments, 1));
}).bind(
(function() {
var _exports = {};
function test (a) {
var s = test2();
alert(s + ' test: ' + a);
}
function test2 () {
return 'test2';
}
_exports['test'] = test;
_exports['test2'] = test2;
return _exports;
})()
)};
Singleton.call('test', 'foo and stuff');
//
previous post:
You are talking about Function#bind functionality that enables 'customizing' function's context. .bind() your call method to required 'local context' like this:
var Singleton = {
//main entry point
// call with fn name, args...
call: (function() {
// here `this` (context) is object bound to `call` method
// not `global` object, which is default for 'unbound' functions
var locals = this; // {fns, shift, loge, isfunc}
var fn;
var fun;
var x;
if (arguments.length == 0)
return;
// get the fn name
fn = locals.shift(arguments);
// '...index into local scope and get a fn'
fun = locals.fns[fn];
// log error if not found
if (!locals.isfunc(fun)) {
locals.loge('Singleton: function not found:' + fn);
return;
}
// ok, run the function
return fun.apply(window, arguments);
// lock `call`'s context to provided object
// and use `this` to reference it inside `call`
}).bind({
fns: (function(_) {
// and you can '...create a library of inter-dependent STATIC functions'
// in this closure and invoke them in `call` method above
_.test = function (a) {
var s = _.test2();
alert(s + ' test: ' + a);
};
_.test2 = function() {
return 'test2';
};
return _;
})({}),
// and create couple of helper methods as well...
isfunc: (function(_getclass) {
_getclass.func = _getclass(_getclass);
return ('function' !== typeof(/foo/)) ?
function(node) {
return 'function' == typeof node;
} :
function(node) {
return _getclass.func === _getclass(node);
};
})(Function.prototype.call.bind(Object.prototype.toString)),
loge: console.warn,
shift: Function.prototype.call.bind(Array.prototype.shift)
}),
};
Singleton.call('test', 'foo and stuff');
// eof

Here's one 'in your face answer', because I really don't like what I see here.
I don't see why you need this kind of construct, you already have that as part of language core.
1. dynamic lookup
you are doing it in a rather 'unprecedented' kind of way,
hashes already do that for you, and it's lightning fast to do a hash search.
If you are eval()-ing random strings to do simple name lookup you really have to
step aside from a keybord for a while... (no offense please)
2. closures
you are saying about 'using closures' which you actualy don't use.
your call function redeclares test functions each time it gets called,
and looks the ('fresh version') functions in it's own variable scope table,
instead of lookig them up in parent scope chains (aka. closures)
outside it's lexical location
3. nfe vs. nfd
ie. named function expressions vs. named function declarations
...you cannot assign a function to a local var and have it retain closure.
It is a feature, you might not be aware of how it works (it tripped me up as well).
check this article out for clarification
4. exceptions
Singleton: function name not found... x4!
Just go ahead and call a function,
interpreter will throw for you anyway if it cannot find/execute
5. eval (aka. ^^)
Singleton.call.ctx.fun = eval(Singleton.call.ctx.fn);
eval takes any string here(#!), and gladly executes ones like:
'for(;;);', or 'while(1);'... forever.
You probably don't want to have any code running unless it was your stuff.
6. arguments handling
It is considered best practice out there to use single (Object) options parameter
to 'fine tune' any significant piece of bundled functionality,
instead of trying to figure that out by type checking provided argument list
Here's, in couple of simple lines, what I (and as I can see #Jimmy Breck-McKye) suggest you should do:
var Singleton.call = (function () {
var funcmap = {
'f_1': function () {},
// etc.
'f_N': function () {},
};
return function (options) {
// options members:
// context, (Object) context, (defaults to global if none is given)
// func, (String) function_name,
// args, (Array) arguments to pass into a function
// this line does everything your 100+ lines long snippet was trying to:
// look's up parent scope for a function, tries to run it
// passing provided data, throws if it gets stuck.
return funcmap[options.func].apply(options.context, options.args);
};
})();
//

Answering my own question here.
The core of the issue is that you cannot assign a function to a local var and have it retain closure.
Consider that when writing a function with global and window scope, this is not necessary to call another function with identical scope. Such is not the case with member functions.
Another way of saying this is that there is no space where your cursor can sit and as you declare a function it automatically gets attached to the current this.
function fn(){}// if we are in global scope, then window.fn becomes defined
// but if we are inside, say, a constructor, simple declaration will not attach
// it to this, but fn is available in scope.
Any assignment on function declaration BREAKS part of the expected closure:
var IdentifierAvailableToClosure=function Unavailable(){}
But assignment after declaration works:
function NowAvailable(){}
var SynonymAvailableToo=NowAvailable;
This is what I meant by not wanting to repeat the name twice to get the mechanism to work.
This fact made me abandon other methods and rely on eval as suggested. Here is a first draft:
// This object is an encapsulation mechanism for a group of
// inter-dependent, static-ish, functions that can call each other
// without a this pointer prefix.
// Calls take the form of:
// Singleton.call(functionName:String [,arg1]...)
// or
// Singleton.call(contextObject:Object, functionName:String [,arg1]...)
// If a context is not provided, window is used.
//
// This type of mechanism is useful when you have defined a group
// of functions in the window/global scope and they are not ready
// to be formalized into a set of classes, or you have no intention
// of doing that
//
// To illustrate the issue, consider that a function
// which is defined in window/global scope
// does not have to use the this pointer to call a function of
// identical scope -- yet in a class member function, the this pointer
// MUST be used
// Therefore, trying to package such functions requires injecting
// the this pointer into function bodies where calls to associater
// functions are made
//
// Usage is primarily for development where one has control over
// global namespace pollution and the mechanism is useful in
// refactoring prior to formalization of methods into classes
var Singleton={
// Main call point
call:function(){
// Bail with error if no args
if (arguments.length==0) {
throw('Singleton: need at least 1 arg');
}
// As all functions in the local scope library below
// have access to the local scope via closure, we want to reduce
// pollution here, so lets attach locals to this call
// function instead of declaring locals
//
// Prepare to call anon fn
Singleton.call.args=arguments;
// Make ctx have args, context object, and function name
Singleton.call.ctx=(function (){// return args,ctx,name
// out
var args=[];
//locals
var x, fn;
// collapse identifier
var a=Singleton.call.args;
// closure object avail to functions, default to window
that=window;
// first real function argument
var arg_start=1;
// first arg must be function name or object
if (typeof a[0]=='string') {// use window ctx
fn=a[0];
// if first arg is object, second is name
}else if (typeof a[0]=='object') {
// assign given context
that=a[0];
// check second arg for string, function name
if (typeof a[1]!='string') {
var err='Singleton: second argument needs to be a fn name'
+' when first arg is a context object';
throw(err)
return;
}
// ok, have a name
fn=a[1];
// args follow
arg_start=2;
}else{
// improper arg types
var err='Singleton: first argument needs to be a string or object';
throw(err)
}
// build args array for function
for (x=arg_start;x<a.length;x++) {
args[args.length]=a[x];
}
// return context
return {
args: args
,that:that
,fn:fn
};
})();
// using function library present in local scope, try to find specified function
try{
Singleton.call.ctx.fun=eval(Singleton.call.ctx.fn);
}catch (e){
console.error('Singleton: function name not found:' + Singleton.call.ctx.fn);
throw('Singleton: function name not found:' + Singleton.call.ctx.fn);
}
// it must be a function
if (typeof Singleton.call.ctx.fun !== 'function') {
console.error('Singleton: function name not found:' + Singleton.call.ctx.fn);
throw('Singleton: function name not found:' + Singleton.call.ctx.fn);
}
// library functions use that instead of this
// that is visible to them due to closure
var that=Singleton.call.ctx.that;
// Do the call!
return Singleton.call.ctx.fun.apply(that, Singleton.call.ctx.args);
//
// cool library of functions below,
// functions see each other through closure and not through this.fn
function test(s){
alert(test2()+' test:'+s);
}
function info_props(){
console.info(this_props());
}
function test2(){
return 'test2';
}
function this_props(){
var s='';
for (var i in that) {
s+=' '+i;
}
return s;
};
}
};

What is the difference in the following JS syntax?

Following are two ways to define BW.Timer. Can someone tell me what the difference is? I am not certain the first is even valid, but if it is valid, what is different about using the myfunc=(function(){}()) syntax?
BW.Timer = (function () {
return {
Add: function (o) {
alert(o);
},
Remove: function (o) {
alert(o);
}
};
} ());
And...
BW.Timer = function () {
return {
Add: function (o) {
alert(o);
},
Remove: function (o) {
alert(o);
}
};
};

The first is the return-value of the immediately-invoked function. The second is a function. It essentially comes down to what the difference is between these:
var f = (function() { return 0; })();
var f = function() { return 0; };
Since the first function is called immediately, the value of 0 is given to the variable f. The first f is not a function. However, the second f we must call in order to get the value:
f(); // 0
It follows that in your example, the first BW.Timer is the object literal itself and the second is a function returning an object literal. You must call the function in order to get to the object:
BW.Timer().Add(x);
Why use the first then?
You might ask yourself why one would use a syntax like a = (function() { return {}; })() instead of a = {}, but there's a good reason. An IIFE (Immeditately-Invoked Function Expression), unlike a regular function allows the emulation of static variables (variables that maintain their value through a single instance). For example:
var module = (function() {
var x = 0;
return { get: function() { return x },
set: function(n) { x = n }
};
})();
The above a text-book example of the Module Pattern. Since the function is called right away, the variable x is instantiated and the return value (the object) is given to module. There's no way we can get to x other than by using the get and set methods provided for us. Therefore, x is static, meaning its variable won't be overridden each time you use module.
module.set(5);
module.get(); // 5
On the other hand, let's see an example where module is declared as a function instead:
// assume module was made as a function
module().set(5);
module().get(); // 0
When we call module() the x variable is overridden each time. So we're effectively using different instances of module and x each time we call module.

The difference is rather large.
In the first case, BW.Timer is executed when it is first encountered, and that is the static version assigned to BW.Timer. In that instance, BW.Timer.Add(1) may be used. Each call to BW.Timer will be the same object.
In the second case, BW.Timer is not executed when first encountered, and instead is a function referece which must be invoked BW.Timer(). For Add to be used, this must be the case BW.Timer().Add(1). Also, you can issue var timer = new BM.Timer();. Each instance of BW.Timer() will be unique here.

In the first example BW.Timer references an object that the self-executing function returns, while in the second example it references a function object, in other words it's a function that can be executed BW.Timer().