To compare arrays, loop through them and compare every value:

Comparing arrays:

// Warn if overriding existing method
if(Array.prototype.equals)
    console.warn("Overriding existing Array.prototype.equals. Possible causes: New API defines the method, there's a framework conflict or you've got double inclusions in your code.");
// attach the .equals method to Array's prototype to call it on any array
Array.prototype.equals = function (array) {
    // if the other array is a falsy value, return
    if (!array)
        return false;
    // if the argument is the same array, we can be sure the contents are same as well
    if(array === this)
        return true;
    // compare lengths - can save a lot of time 
    if (this.length != array.length)
        return false;

    for (var i = 0, l=this.length; i < l; i++) {
        // Check if we have nested arrays
        if (this[i] instanceof Array && array[i] instanceof Array) {
            // recurse into the nested arrays
            if (!this[i].equals(array[i]))
                return false;       
        }           
        else if (this[i] != array[i]) { 
            // Warning - two different object instances will never be equal: {x:20} != {x:20}
            return false;   
        }           
    }       
    return true;
}
// Hide method from for-in loops
Object.defineProperty(Array.prototype, "equals", {enumerable: false});

Usage:

[1, 2, [3, 4]].equals([1, 2, [3, 2]]) === false;
[1, "2,3"].equals([1, 2, 3]) === false;
[1, 2, [3, 4]].equals([1, 2, [3, 4]]) === true;
[1, 2, 1, 2].equals([1, 2, 1, 2]) === true;

You may say "But it is much faster to compare strings - no loops..." well, then you should note there ARE loops. First recursive loop that converts Array to string and second, that compares two strings. So this method is faster than use of string.

I believe that larger amounts of data should be always stored in arrays, not in objects. However if you use objects, they can be partially compared too.
Here's how:

Comparing objects:

I've stated above, that two object instances will never be equal, even if they contain same data at the moment:

({a:1, foo:"bar", numberOfTheBeast: 666}) == ({a:1, foo:"bar", numberOfTheBeast: 666})  //false

This has a reason, since there may be, for example private variables within objects.

However, if you just use object structure to contain data, comparing is still possible:

Object.prototype.equals = function(object2) {
    //For the first loop, we only check for types
    for (propName in this) {
        //Check for inherited methods and properties - like .equals itself
        //https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/hasOwnProperty
        //Return false if the return value is different
        if (this.hasOwnProperty(propName) != object2.hasOwnProperty(propName)) {
            return false;
        }
        //Check instance type
        else if (typeof this[propName] != typeof object2[propName]) {
            //Different types => not equal
            return false;
        }
    }
    //Now a deeper check using other objects property names
    for(propName in object2) {
        //We must check instances anyway, there may be a property that only exists in object2
            //I wonder, if remembering the checked values from the first loop would be faster or not 
        if (this.hasOwnProperty(propName) != object2.hasOwnProperty(propName)) {
            return false;
        }
        else if (typeof this[propName] != typeof object2[propName]) {
            return false;
        }
        //If the property is inherited, do not check any more (it must be equa if both objects inherit it)
        if(!this.hasOwnProperty(propName))
          continue;
        
        //Now the detail check and recursion
        
        //This returns the script back to the array comparing
        /**REQUIRES Array.equals**/
        if (this[propName] instanceof Array && object2[propName] instanceof Array) {
                   // recurse into the nested arrays
           if (!this[propName].equals(object2[propName]))
                        return false;
        }
        else if (this[propName] instanceof Object && object2[propName] instanceof Object) {
                   // recurse into another objects
                   //console.log("Recursing to compare ", this[propName],"with",object2[propName], " both named \""+propName+"\"");
           if (!this[propName].equals(object2[propName]))
                        return false;
        }
        //Normal value comparison for strings and numbers
        else if(this[propName] != object2[propName]) {
           return false;
        }
    }
    //If everything passed, let's say YES
    return true;
}  

However, remember that this one is to serve in comparing JSON like data, not class instances and other stuff. If you want to compare more complicated objects, look at this answer and it's super long function.
To make this work with Array.equals you must edit the original function a little bit:

...
    // Check if we have nested arrays
    if (this[i] instanceof Array && array[i] instanceof Array) {
        // recurse into the nested arrays
        if (!this[i].equals(array[i]))
            return false;
    }
    /**REQUIRES OBJECT COMPARE**/
    else if (this[i] instanceof Object && array[i] instanceof Object) {
        // recurse into another objects
        //console.log("Recursing to compare ", this[propName],"with",object2[propName], " both named \""+propName+"\"");
        if (!this[i].equals(array[i]))
            return false;
        }
    else if (this[i] != array[i]) {
...

I made a little test tool for both of the functions.

Bonus: Nested arrays with indexOf and contains

Samy Bencherif has prepared useful functions for the case you're searching for a specific object in nested arrays, which are available here: https://jsfiddle.net/SamyBencherif/8352y6yw/