If I have a 45% win rate with hero X, and a 60% win rate with hero Y, does that mean I'm better with hero Y? Does it tell me anything?

Stats are a useful tool but you have to be careful not to read too much into them. I have a 70%+ winrate on one hero that I know that I am not particularly good at playing. But it's the hero I tend to play when my team is already doing really well or I am on a very good team. One of my best characters has a very low winrate because it is my "tilt" character -- the hero I switch to when the match is probably already lost and my team is not coordinating. The end result of both of these behaviors is some pretty skewed stats.

Since the time of this post, a majority of stats in the Career Profile now calculate based on time played every ten minutes. For hero specific win rates, this means for every ten minutes of gameplay in a match you won will count toward a win for that hero (and the same with draws and losses). This makes the win rates a little more accurate but Jeff Kaplan is still correct that your actual hero specific win rates are influenced by how the match is playing out.