Same maneuver, different names. Flare is more US, Round-out more British, I think. Both mean the same thing - change aircraft attitude from approach to landing; this reduces sink rate and sets up for the specific type of landing touchdown being done: fully-held off low energy (tail and main at same time) or flown on main only (Blanik, Schweizers). Tailwheel airplanes can do 3-point, wheel landings, or one main wheel crosswind landings. And in an F-4 or F-18 you don't even have to bother, just drive it onto the runway (or, as my navy friends used to say back before PC: "Flare to land, squat to pee"). In my LS6 I can get the tail wheel to touch just before the main if really going for min energy; a "one wheel" landing?

And "flair" is how stylishly you accomplish the "flare".

66