CAT5e cable meets spec for gigabit (as in fact, does CAT5) - assuming your installation meets specification, you will see no change in performance by switching to CAT6.
The physical difference between CAT5, CAT5e & CAT6 is the number of twists per inch - the "tighter" the twist, the better the noise rejection of the cable - so CAT6 is no "faster" - it's less noisy.