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.