Alright so I don't play a lot of poker, but I'm still going to attempt to answer this. Feel free to correct me though.

Assuming you 2 hearts in your hand, and two more come out on the flop, you then need a heart to come on either the turn or the river.

Assuming that you don't know any of the other players cards:

The odds of a heart coming out on the turn are 9/47 or 19.1%

The odds of a heart coming out on the river are 9/46 or 19.6%

So the odds of hitting a flush when you have 4 to a flush after the flop are 19.1%+19.6% or 38.7%

So it would seem that your chances of winning are somewhere around 40%, although it depends a lot on which exactly cards you and your opponent have. (If one of his aces is in the suit of your flush draw, that hurts your chances even more.)

I also simulated this situation on this website:
http://poker-tools.flopturnriver.com/Poker-Odds-Calculator.php
and again found that the odds were about 60-40 in favor of the pocket aces.

However this is all assuming that both players stay in till the end of the hand. I have heard people say that what your cards are isn't nearly as important as how you bet them. And if you do hit the flush, your payoff will most likely be bigger than if the guy with the aces wins, so there is a lot to consider.

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