SNES Final Fight has a lot of differences between the Japanese and American versions. Generally speaking, changes made to the American version were also made to the European version, apart from a few specific things which are highlighted below.

One little extra thing about the European version: in the sound test, you can play track 0A. In the Japanese and American versions however, it can't. This track is not used in the game at all. It would have meant to have been used in the Industrial Area's elevator ride, had that level not been cut from this version. I have no idea why the European version lets you play this track.

Japanese	American	European
The Capcom logo for the European version is different to the other versions, as it fades in and then out, using the usual jingle.

Japanese	American	European
The title screens between versions are also slightly different.

Japanese	American
The enemies Dug and Simons had dark skin in the Japanese release, but in the American version they have lighter skin colors. Also I cannot believe the censors missed the fact that it clearly says "DICK" on the door.

Japanese	American
Poison and Roxy are in the Japanese version, but were replaced with Billy and Sid in the other versions. The replacements look different, but have the same moves.

Japanese	American
When you hit someone with a normal attack in any version, you get a small explosion effect. However, in the Japanese version, when you hit an enemy with a weapon, you will see a little blood squirt, like in the arcade game. This was dropped from the American version, and instead you just have the explosion effect again.

Japanese	American
Whilst Damnd kept his arcade name in the Japanese version, it was changed to "Thrasher" in America.

Japanese	American
In the Japanese version, the "OH! MY GOD" line was kept for the first bonus game. In the American version, it was changed to "OH! MY CAR ".

Japanese	American
Sodom managed to keep his arcade name in the Japanese version, but in the American version it was changed to "Katana". This was because his original name guards relation to the biblical cities of Sodom and Gomorrah, or to sodomy.

Japanese	American
In the West Side level, the sign outside the door is different depending on the version. The Japanese version has "BAR", whilst the American version has "CLUB".

Japanese	American	European
In West Side's bar/club, there is a layering error in both the Japanese and American versions which prevents a group of people from appearing at the bar. This was fixed for the European version, and was also fixed in Final Fight Guy.

Japanese	American	European
Later on in West Side, there is anohter layering error which stops a couple of people appearing in the Japanese and American versions. This was again fixed in the European version, along with Final Fight Guy.

Japanese	American
The statues you can often see in the Up Town level are also different. In the Japanese version, the breasts are more exposed, whilst in the American version they have been covered up significantly.

Japanese	American
Here is the strangest change in the entire game. For some reason, Belger's wheelchair is completely different between versions. The Japanese version has Belger's chair looking like it does in the arcade version. However, in the American version the chair was changed. The best idea I've heard for this change is that somebody thought Belger was disabled, and objected to having to fight him. So the chair was changed from a wheel chair to a executive chair with small wheels underneath.