I don't personally think it's an *offensive* term Artbrooks. I just *personally* dislike it. The term has some decidedly negative connotations especially from Europe, because of the type of behavior commonly to be seen among classic self-defined "Brits" abroad - particularly those of the lager-swilling shaven-headed variety.

However this is not the only usage. As a diminuitive of "British", we can commonly see more 'affectionate' or informal usage of the term too. And I expect that's precisely the way most Americans here would use it.