Why does different have to be separate?
It shouldn't be separate, nor should any differences be the first consideration.
There is quite a bit more to black history than just slavery, go read.
European American = white, its the same thing. African-American = black, once again same thing. Irish American, except for a brief period of discrimination of them as immigrants, they are now and have been for a while, white. Hispanic, Asian, etc. we all have different sub-histories, and I think that should be recognized.
The Spanish Moors were certainly white, nor are all people of African descent black. Do you think white South Africans or Egyptians consider themselves "African American?" And what of people with African or Asian heritage who are born in France or England and consider themselves to be French or British? They sure aren't white but they consider themselves European. It is misleading and can be insulting to attribute a race to a particular continent.
Why does being different drive a wedge? I mean, you can say whatever you want, but I am different from you, and there is nothing wrong with that. There is no need to cover up being different.
Being different does not drive a wedge. Demanding deference or acknowledgement or privileges or special considerations because of one's differences drives wedges; and when you lead with the 'difference' and make that more important than the whole group, that drives a wedge.
The term African-American denotes someone of african descent. That does not mean it has anything to do with some sort of desire for africa over America. America is supposed to tolerate diversity, once again, being different is not the same as separate, if you choose to separate yourself b/c you or someone else is different, that is a personal decision.
The term African-American should denote somebody who was born in Africa and has immigrated to America. It should denote nothing else. As already explained, it cannot and should not denote race.
Lets make an analogy. Christian : American as Baptist : African-American/black. Baptists, Methodists, Lutherans, etc. all believe in the Bible, but they are somewhat different, so they call themselves different. But most would agree that they are all part of the Christian community. In the same way, blacks, whites, hispanics, asians, etc. are different, and so call themselves differently, but they are still the makeup of the American community.