Except Britain only contain on country, so British is pretty clear. We'll get to annoy them once (if) Scotland gain independence.
Define: Country
Because depending on how you define it the UK has 4. Though really they're more like meta-counties.
Also Great Britain doesn't include NI, which is a part of the UK.
The United Kingdom contains four countries: England, Wales, Scotland, and Northern Ireland. Great Britain describes the first three. It is, in effect, a country composed of several countries. Hence, someone from England can introduce themselves as both English and British, while a Scot could introduce themselves as Scottish and British, etc.
The U.S. doesn't get the same deal because we're a federalist nation; the states are subordinate territorial units with a certain degree of self-governance; someone might introduce themselves as, say, a Kentuckian when speaking to other
Americans, but probably not to people from other countries, because there is ultimately no national identity associated with the states. Except for remnants of the Republic of Texas and the Confederacy, but we prefer not to talk about those.
"American" is a useful term because it would be pants-on-head retarded to call ourselves United-States-of-Americans every time we mentioned our nationality. It isn't used to try to diminish the importance of Canada, Mexico, the Caribbean, or Central and South America, but because it's the closest we have to a good name for our nationality.
Someone from the Republic of Ireland doesn't call themselves Republic-of-Irish, because that would be
stupid, even if it would be more accurate. People from the Republic of Korea (or the DPRK) aren't Republic-of-Koreans (or Democratic-Peoples'-Republic-of-Koreans) because it's a ponderous mouthful that isn't necessary. Same deal here.