I'm not 100% sure about Britain - I've heard them called Black British or Caribbean British (because so many Blacks in Britain are from there).
African-American (formerly Afro-American, formerly Black, formerly Colored, formerly N e g r o) I think came about as a term in the 60s and the 70s. At that time, many Black people were finding ways to identify themselves for themselves - instead of being defined by other people. Before that, often Black people were referred to by names that many white people gave them - like n***** (which some black people use today - I personally don't agree with that), coon, darky, j u n g l e b u n n y, spot, night, *******, octaroon, quardroon, and many, many other - mostly derogatory - terms.
Personally, I don't see why other people are so up in arms about it. If White people feel more comfortable calling themselves White, then great. No one tells White people that they must call themselves German-American or anything else - so why should Black people be any different? (Although I've heard lots of Italians use the hyphenation in a heartbeat - Italian-Americans). I think that each group should have the right to determine what they want to be called. Remember when women wanted to be called Ms. instead of Miss or automatically being called Mrs.? There was a statement that they wanted to make - that they would not be defined by their marital status - and the women who wanted to use that were free to do so. I don't see where this is really that different. I personally use African-American and Black interchangeably. I also use Ms. instead of Miss - but I don't get upset if someone calls me Miss (after all - I certainly don't mind being mistaken for someone younger!)
There are other groups in this country that hyphenate as well - think about Native-Americans, Asian-Americans (remember when Asians were called Orientals?), Hispanic-Americans. I think it is interesting that people raise the question about African-Americans, but not these other groups. American is still in the name - and the other part of the name is an adjective that modifies American - which is the noun and carries the weight of the description. No one is saying American-Africans - which would have a very different connotation.
I think that America has always labeled its citizens and that there has always been racial profiling. Even if tomorrow we renamed racial groups as Daisy, Gardenia, Rose, and Lily - or did away with the names altogether there'd probably still be an issue - because racism (and sexism) is not really a matter of language only - but of the heart. We haven't dealt with race collectively as a country - not really - although we came close to doing so in the 60s and 70s. So, like anything else that is repressed it festers. And out of the heart, flows the language that we use.
I agree with you that America is really into categories - I'm not sure why that is - but I think it puts people in artificial boxes. If you travel abroad - then people put you in different boxes - and it can really blow your mind. It also shows you how arbitrary this race stuff really is. But, I'll be very honest with you. In my experience of traveling abroad, when I meet a White American - seven times out of ten they will not see or treat me as a fellow American - even though the people whose country I'm visiting do. While it is difficult because I might really hunger for American things - it works in my favor because it pushes me a lot closer to the people that I'm visiting - and generally I'm able to form really interesting relationships with them because I end up immersing myself in the culture a bit more.
But, even with all of this, I'm proud to be an American - whether I call myself African-American, Black, or simply American. And just about every other Black person that I know feels exactly the same way.