The Date object will do what you want - construct one for each date, then compare them using the >, <, <= or >=.

The ==, !=, ===, and !== operators require you to use date.getTime() as in

var d1 = new Date();
var d2 = new Date(d1);
var same = d1.getTime() === d2.getTime();
var notSame = d1.getTime() !== d2.getTime();

to be clear just checking for equality directly with the date objects won't work

var d1 = new Date();
var d2 = new Date(d1);

console.log(d1 == d2);   // prints false (wrong!) 
console.log(d1 === d2);  // prints false (wrong!)
console.log(d1 != d2);   // prints true  (wrong!)
console.log(d1 !== d2);  // prints true  (wrong!)
console.log(d1.getTime() === d2.getTime()); // prints true (correct)

I suggest you use drop-downs or some similar constrained form of date entry rather than text boxes, though, lest you find yourself in input validation hell.

For the curious, date.getTime() documentation:

Returns the numeric value of the specified date as the number of milliseconds since January 1, 1970, 00:00:00 UTC. (Negative values are returned for prior times.)

if (date1.getTime() > date2.getTime()) {
    alert("The first date is after the second date!");
}

Reference to Date object