I can't speak knowledgeably about AE use, but I think of "cigarette stub" as BE, so a BE tag seems appropriate.
I was also surprised at how common butt seems to have become. I have definitely seen it, but I would have thought it was still less common. I may be more conscious than the average Brit that butt is also used in the US for arse/bottom; I suppose that might explain why I tend to think that butt is not used in the UK.
I would, personally, expect to see "cigarette stub" in a law enforcement (litter) context. I would also expect "Please do not throw cigarette stubs in the urinal, as it makes them soggy and hard to light.", but I can imagine seeing it replaced by cigarette "butts".
I think of Andy Capp having a fag-end hanging from his lip.
I wonder if usage in Britain has changed with the demise of non-filter cigarettes