The thing that goes up and down with horses and such is a merry-go-round or a carousel. They are the same -- at least in American English. There is something else called a merry-go-round that you might see at a children's playground where they spin it themselves. I have never heard that called a carousel. Kids usually just call that a merry-go-round.

In U.S. English I am not aware of ANY difference in meaning between a carousel and a merry-go-round. Carousel is becoming more common, because it sounds fancier and more dignified. Thus, there are (wealthy) people who collect "carousel horses," not "merry-go-round horses," and in public parks the ride will officially be called, for example, the Greenway Carousel in Boston--but people will ask their kids if they want to ride "the merry-go-round." I agree with Kevin about the playground ride. A bit of outdated trivia. Kodak developed an extreme successful slide projector, that became almost universal from perhaps 1960 through 2000 for showing pictures at talks, presentations, and lectures. Their trade name for it was "carousel," because it used a well-engineered circular tray that held 80 slides and fed them reliably and quickly, without jamming. So you will occasionally see references to "a carousel" meaning "a slide tray."

Merry-go-round always refers to the ride, and as far as I know it is used in all English dialects. Carousel is the more general concept of a disc or cylinder (something round) that rotates, so a merry-go-round is a kind of carousel. On the web, pictures that change continuously are said to be "rotating" or "in rotation" so we call that a carousel, too. If the context is a carnival or something like that, the word carousel is well understood to be a merry-go-round.

None - they are the same. ( and I have been a fan of them my entire life. I began riding them in 1966 and in 2016 I intend to be on one again ! )

Nothing. They are synonyms. The most commonly used in spoken English? It will differ between regions. It's not a word that comes up particularly often but I personally I would call it a 'merry-go-round.' .