I would say that they are basically the same, but not exactly the same. "Here I come" is more demonstrative, and is more of an exclamation than merely saying, "I'm coming". There are situations where the one version is more suitable than the other. For example: You are standing in the airport, waiting to board your flight to the U.S.A., and you say, "America, here I come". I don't think many people will say, "America, I'm coming".