Javascript objects are only equal if they refer to the same object-reference. Which is not the case in your code.
(reference: What is the standard definition of object equality for "=="?)
Edit: With a little type casting you can do:
var str="Oct 13,2011";
var date1 = new Date(str);
var date2 = new Date(str);
console.log(+date1 == +date2); // true