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The bracket notation allows you to access properties by name stored in a variable:

var obj = "abc" : "hello" ;
var x = "abc";
var y = obj[x];
console.log(y); //output - hello

obj.x would not work in this case.

Dot notation does not work with some keywords (like new and class) in internet explorer 8.

I had this code:

//app.users is a hash
app.users.new = 
 // some code

And this triggers the dreaded "expected indentifier" (at least on IE8 on windows xp, I havn't tried other environments). The simple fix for that is to switch to bracket notation:

app.users['new'] = 
 // some code

Be careful while using these notations:
For eg. if we want to access a function present in the parent of a window.
In IE :

window['parent']['func']

is not equivalent to

window.['parent.func']

We may either use:

window['parent']['func'] 

or

window.parent.func 

to access it

The two most common ways to access properties in JavaScript are with a dot and with square brackets. Both value.x and value[x] access a property on value—but not necessarily the same property. The difference is in how x is interpreted. When using a dot, the part after the dot must be a valid variable name, and it directly names the property. When using square brackets, the expression between the brackets is evaluated to get the property name. Whereas value.x fetches the property of value named “x”, value[x] tries to evaluate the expression x and uses the result as the property name.

So if you know that the property you are interested in is called “length”, you say value.length. If you want to extract the property named by the value held in the variable i, you say value[i]. And because property names can be any string, if you want to access a property named “2” or “John Doe”, you must use square brackets: value[2] or value["John Doe"]. This is the case even though you know the precise name of the property in advance, because neither “2” nor “John Doe” is a valid variable name and so cannot be accessed through dot notation.

In case of Arrays

The elements in an array are stored in properties. Because the names of these properties are numbers and we often need to get their name from a variable, we have to use the bracket syntax to access them. The length property of an array tells us how many elements it contains. This property name is a valid variable name, and we know its name in advance, so to find the length of an array, you typically write array.length because that is easier to write than array["length"].

Generally speaking, they do the same job.

Nevertheless, the bracket notation gives you the opportunity to do stuff that you can't do with dot notation, like

var x = elem["foo"]; // can't do elem.foo;

This can be extended to any property containing special characters.

You need to use brackets if the property names has special characters:

var foo = 
 "Hello, world!": true,

foo["Hello, world!"] = false;

Other than that, I suppose it's just a matter of taste. IMHO, the dot notation is shorter and it makes it more obvious that it's a property rather than an array element (although of course JavaScript does not have associative arrays anyway).

Bracket notation can use variables, so it is useful in two instances where dot notation will not work:

1) When the property names are dynamically determined (when the exact names are not known until runtime).

2) When using a for..in loop to go through all the properties of an object.

source: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Working_with_Objects

You have to use square bracket notation when -

1. 
   

   The property name is number.

   
   
   

   var ob = 
    1: 'One',
    7 : 'Seven'
   
   ob.7 // SyntaxError
   ob[7] // "Seven"
   

   
   


2. 
   

   The property name has special character.

   
   
   

   var ob = 
    'This is one': 1,
    'This is seven': 7,
    
   ob.'This is one' // SyntaxError
   ob['This is one'] // 1
   

   
   


3. 
   

   The property name is assigned to a variable and you want to access the
   property value by this variable.

   
   
   

   var ob = 
    'One': 1,
    'Seven': 7,
   
   
   var _Seven = 'Seven';
   ob._Seven // undefined
   ob[_Seven] // 7
   

   
   

Case where notation is helpful :

If your object is dynamic and there could be some random values in keys like number and or any other special character, for example -

var a = 1 : 3 ;

Now if you try to access in like a.1 it will through an error, because it is expecting an string over there.

Let me add some more use case of the square-bracket notation. If you want to access a property say x-proxy in a object, then - will be interpreted wrongly. Their are some other cases too like space, dot, etc., where dot operation will not help you. Also if u have the key in a variable then only way to access the value of the key in a object is by bracket notation. Hope you get some more context.