(Sourced from here.)

Square bracket notation allows the use of characters that can’t be used with dot notation:

var foo = myForm.foo[]; // incorrect syntax
var foo = myForm["foo[]"]; // correct syntax

including non-ASCII (UTF-8) characters, as in myForm["ダ"] (more examples).

Secondly, square bracket notation is useful when dealing with
property names which vary in a predictable way:

for (var i = 0; i < 10; i++) {
  someFunction(myForm["myControlNumber" + i]);
}

Roundup:

• Dot notation is faster to write and clearer to read.
• Square bracket notation allows access to properties containing
  special characters and selection of
  properties using variables

Another example of characters that can’t be used with dot notation is property names that themselves contain a dot.

For example a json response could contain a property called bar.Baz.

var foo = myResponse.bar.Baz; // incorrect syntax
var foo = myResponse["bar.Baz"]; // correct syntax

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.

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"].

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
}

Both foo.bar and foo["bar"] access a property on foo but not necessarily the same property. The difference is in how bar is interpreted. When using a dot, the word after the dot is the literal name of the property. When using square brackets, the expression between the brackets is evaluated to get the property name. Whereas foo.bar fetches the
property of value named “bar” , foo["bar"] tries to evaluate the expression "bar" and uses the result, converted to a string, as the property name.

Dot Notation’s Limitation

If we take this object:

const obj = {
  123: 'digit',
  123name: 'start with digit',
  name123: 'does not start with digit',
  $name: '$ sign',
  name-123: 'hyphen',
  NAME: 'upper case',
  name: 'lower case'
};

accessing its properties using dot notation

obj.123;      // ❌ SyntaxError
obj.123name;  // ❌ SyntaxError
obj.name123;  // ✅ 'does not start with digit'
obj.$name;    // ✅  '$ sign'
obj.name-123;  // ❌ SyntaxError
obj.'name-123';// ❌ SyntaxError
obj.NAME; // ✅ 'upper case'
obj.name; // ✅ 'lower case'

But none of this is a problem for the Bracket Notation:

obj['123'];     // ✅ 'digit'
obj['123name']; // ✅ 'start with digit'
obj['name123']; // ✅ 'does not start with digit'
obj['$name'];   // ✅ '$ sign'
obj['name-123']; // ✅ 'does not start with digit'
obj['NAME']; // ✅ 'upper case'
obj['name']; // ✅ 'lower case'

Accessing variable using variable:

const variable = 'name';
const obj = {
  name: 'value'
};
// Bracket Notation
obj[variable]; // ✅ 'value'
// Dot Notation
obj.variable; // undefined

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 name 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).

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

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
   

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

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.

Dot notation is always preferable. If you are using some “smarter” IDE or text editor, it will show undefined names from that object.
Use brackets notation only when you have the name with like dashes or something similar invalid. And also if the name is stored in a variable.

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.

An example where the dot notation fails

json = { 
   "value:":4,
   'help"':2,
   "hello'":32,
   "data+":2,
   "😎":'😴',
   "a[]":[ 
      2,
      2
   ]
};

// correct
console.log(json['value:']);
console.log(json['help"']);
console.log(json["help""]);
console.log(json['hello'']);
console.log(json["hello'"]);
console.log(json["data+"]);
console.log(json["😎"]);
console.log(json["a[]"]);

// wrong
console.log(json.value:);
console.log(json.help");
console.log(json.hello');
console.log(json.data+);
console.log(json.😎);
console.log(json.a[]);

The property names shouldn’t interfere with the syntax rules of javascript for you to be able to access them as json.property_name

Or when you want to dynamically change the classList action for an element:

// Correct

showModal.forEach(node => {
  node.addEventListener(
    'click',
    () => {
      changeClass(findHidden, 'remove'); // Correct
    },
    true
  );
});

//correct
function changeClass(findHidden, className) {
  for (let item of findHidden) {
    console.log(item.classList[className]('hidden'));// Correct
  }
}

// Incorrect 
function changeClass(findHidden, className) {
  for (let item of findHidden) {
    console.log(item.classList.className('hidden')); // Doesn't work
  }
}

The dot notation and bracket notation both are used to access the object properties in JavaScript. The dot notation is mostly used as it is easier to read and comprehend. So why should we use bracket notation and what is the difference between then? well, the bracket notation [] allows us to access object properties using variables because it converts the expression inside the square brackets to a string.

const person = {
  name: 'John',
  age: 30
};

//dot notation
const nameDot = person.name;
console.log(nameDot);
// 'John'

const nameBracket = person['name'];
console.log(nameBracket);
// 'John'

Now, let’s look at a variable example:

const person = {
  name: 'John',
  age: 30
};

const myName = 'name';
console.log(person[myName]);
// 'John'

Another advantage is that dot notation only contain be alphanumeric (and _ and $) so for instance, if you want to access an object like the one below (contains ‘-‘, you must use the bracket notation for that)

const person = {
  'my-name' : 'John'
}

console.log(person['my-name']); // 'John'
// console.log(person.my-name); // Error