There is a better way using ES7:

Intersection

 let intersection = arr1.filter(x => arr2.includes(x));

For [1,2,3] [2,3] it will yield [2,3]. On the other hand, for [1,2,3] [2,3,5] will return the same thing.

Difference

let difference = arr1.filter(x => !arr2.includes(x));

For [1,2,3] [2,3] it will yield [1]. On the other hand, for [1,2,3] [2,3,5] will return the same thing.

For a symmetric difference, you can do:

let difference = arr1
                 .filter(x => !arr2.includes(x))
                 .concat(arr2.filter(x => !arr1.includes(x)));

This way, you will get an array containing all the elements of arr1 that are not in arr2 and vice-versa

As @Joshaven Potter pointed out on his answer, you can add this to Array.prototype so it can be used like this:

Array.prototype.diff = function(arr2) { return this.filter(x => !arr2.includes(x)); }
[1, 2, 3].diff([2, 3])

Array.prototype.diff = function(a) {
    return this.filter(function(i) {return a.indexOf(i) < 0;});
};

//////////////
// Examples //
//////////////

const dif1 = [1,2,3,4,5,6].diff( [3,4,5] );  
console.log(dif1); // => [1, 2, 6]


const dif2 = ["test1", "test2","test3","test4","test5","test6"].diff(["test1","test2","test3","test4"]);  
console.log(dif2); // => ["test5", "test6"]

Note .indexOf() and .filter() are not available before IE9.

This is by far the easiest way to get exactly the result you are looking for, using jQuery:

var diff = $(old_array).not(new_array).get();

diff now contains what was in old_array that is not in new_array

The difference method in Underscore (or its drop-in replacement, Lo-Dash) can do this too:

(R)eturns the values from array that are not present in the other arrays

_.difference([1, 2, 3, 4, 5], [5, 2, 10]);
=> [1, 3, 4]

As with any Underscore function, you could also use it in a more object-oriented style:

_([1, 2, 3, 4, 5]).difference([5, 2, 10]);

Plain JavaScript

There are two possible intepretations for "difference". I'll let you choose which one you want. Say you have:

var a1 = ['a', 'b'     ];
var a2 = [     'b', 'c'];

1. If you want to get ['a'], use this function:

   function difference(a1, a2) {
     var result = [];
     for (var i = 0; i < a1.length; i++) {
       if (a2.indexOf(a1[i]) === -1) {
         result.push(a1[i]);
       }
     }
     return result;
   }
   

2. If you want to get ['a', 'c'] (all elements contained in either a1 or a2, but not both -- the so-called symmetric difference), use this function:

   function symmetricDifference(a1, a2) {
     var result = [];
     for (var i = 0; i < a1.length; i++) {
       if (a2.indexOf(a1[i]) === -1) {
         result.push(a1[i]);
       }
     }
     for (i = 0; i < a2.length; i++) {
       if (a1.indexOf(a2[i]) === -1) {
         result.push(a2[i]);
       }
     }
     return result;
   }