British English might have continental Europe influences there whereas American English doesn't? Dunno, don't have an explanation for the difference.

As for the "flight", I've always wondered that myself, but never bothered to googled it. Simply assumed it was used for both. Just googled it now, and the consensus seems to be that a flight is an uninterrupted row of stairs. So if you have one of those spiraling staircases and it doesn't stop for 200 steps, that's one flight of stairs. If you have those zig zagging steps that you usually find in modern buildings, even tho there's only one floor between them, if there's a platform in between, that's 2 flight of stairs. So... There you go.