There are relatively few differences between the Japanese and American versions of arcade Final Fight. This would not be the case for certain console ports, however...

Japanese	American
The Japanese version states the exact year when the game takes place: 1989, which happens to be the same year the game was made. The American version changed this to a vague "sometime in the 1990's..." instead. The date Final Fight takes place was later retconned to 1987.

Japanese	American
During the intro, when Haggar turns on his TV, Jessica briefly appears. After a few seconds Damnd will then appear. For the American version, Jessica never appears on the screen, only Damnd does.

Japanese	American
If you get the special ending, the Japanese version has some comments from the developers. The American version replaces them with a single screen of text stating "MY NAME IS POM". Pom refers to Akira Nishitani, the planner for Final Fight. The second screen features comments from Akiman (Akira Yasuda), which was cut entirely from the American version.

Japanese	American
The developer messages were also cut from the American version, replaced with them just stating their nicknames.

Japanese	American
The Japanese version has an additional developer comment (he just says "bird"), which was cut and replaced with Akiman.