I want to get better at chess. I prefer to play online chess, correspondence chess, but I wonder is there any difference between the two as far as improving?

Over-the-board is the best. It's like what turn-based chess and live chess, point to. I love turn-based. But I have to play live every once in a while, even if I don't want to. If I begin to suck at live, I devote more time to it. If I begin to suck at speed chess, live, as opposed to standard live, then I go through a period of playing a lot of speed chess. Internet chess makes it much easier to remain sharp, so my tiny opinion is turn-based is on equal ground with live chess

One thing I try to do is replay the game online from the beginning, trying to see patterns and such. 

Slower time controls are better to improve your game.  The more you think about the position and apply what you do know, the more you can discover the features you don't understand.  Playing blitz is fun, but it takes much longer to get better that way, if you can at all.