I'm not a huge fan of beat-'em-ups, but I do feel compelled to check out the most famous examples of the genre. I've heard Final Fight be mentioned as a classic, so I tried it out on SNES.
It seems like the game is remembered for several reasons. One is that it was one of the launch titles for the SNES, and its arcade-like graphics no doubt were impressive for the time. Another is that its popularity fueled the development of the classic fighting game Street Fighter II, which was definitely an important moment in video game history.
As for the game itself, it seems like fans of the game must be viewing the game through rose-tinted glasses, as I found it to be a typical and unremarkable example of the genre. The controls feel pretty stiff, and even though there are two characters to choose from, the movesets are pretty limited. Most of the time you'll end up spamming the jump attack since it's one of the more powerful moves. The regular punches work okay, and to avoid getting caught by attacks from both directions you can use a suplex move that will move one enemy to the other side of the screen. You also can use a super move that does more damage but costs part of your health, which is a game mechanic shared by other games that originated in the arcades and is always extremely lame. It's too easy to go from being at full health to zero, either by being repeatedly hit by enemies with no chance to recover or by being hit by one of the boss characters who do massive amounts of damage.
The game lacks the third character and additional stage of the original arcade version, but the lack of the co-op mode is what really limits the appeal of the game. The presentation and music are good, although there's not much variety in the settings or enemies. (The game is also an interesting example of censorship that was commonplace at the time.)
This definitely isn't one of the better beat-'em-ups I've played, and it's overly difficult even when using the cheat to adjust the difficulty and to up the number of lives. I might check out its sequel at some point since that game does have a co-op mode and is generally regarded as being better than the first game, but not anytime soon.