If Final Fight retained everything on the Consoles AND Capcom advertised the hell out of it back in the day, then Final Fight would have easily beaten anything SEGA could come up with.
But because SEGA endured making the games better every release (except 3, because they thought we loved really really hard games all of a sudden), Streets of Rage as an overall package is better, as it tried different Enemy Types, Different Locales and always tried to change up mechanics (as basic as they are now, you can still appreciate where they went with it).
Having an amazing Composer does wonders as well, who was in the know about Music of the time (Techno). It holds up today because of the changes and improvements SEGA did over Final Fight and didn't stick to 3 guys saving one person, they decided to have 3 Undercover Cops (all looking and playing differently) try to take down a large gang because the Cops were useless (funny given the Special Attack), and offered the Co-Op option as well as an additional Co-Op attack to assist in clearing enemies....as WELL as offering a "Bad" ending should you go for it.
Although Final Fight introduced a lot of great gameplay mechanics that SOR would eventually use and improved upon, it never really tried to "innovate" after the first game and stuck to what worked in the Arcades of the day, on top of Capcom's priorities of making Street Fighter their bread and butter on the SNES. SEGA managed to not only copy the formula, but decided to improve upon it and advertise that their game is THE game to play on a Mega Drive over the SNES, as well as display that SEGA flair they had popularised in the 90s.
Nostalgia is a huge part to why SoR is still seen as King of the Beat em Ups, but I would say that the strong advertising, willing to improve and innovate the genre early on, as well as create a unique world with a host of strange enemies and characters (akin to The Warriors), the trilogy has held up very well in comparison, as well as the music the games are known for.
If Final Fight tried to do all of this early on, it would have had a chance to be a true rival to Streets of Rage, but because Capcom changed priorities and focused on Street Fighter (which is understandable because look at how much money they made on SF2 and the varients), Final Fight became more as a novelty in the Arcades that you play rather than a real "Must have" Console buy.
I have played both (FF about 12 years ago), and they don't compare. Sure the Sprites in FF are really detailed and large, but I think that was the problem for Console gaming. You need the right sized sprites to take in the World, the Characters, Enemies and Items in the game to make it a truly great experience. SEGA improved in that aspect after the first game (admittedly, I find SoR 1's spites too small), and balanced it out on a Console to make it feel like an Arcade game fit for a Console.
The Mega Drive's CPU would have certainly helped as well, making the action feel faster on many difficulties and adding more enemies in addition to changing how they act depending on if you play on 1 Player or 2. (try playing Mania Mode with 2 Characters but letting one die to challenge it all by yourself).
