I'll throw a few more out there, non-beat 'em up this time:
Avenging Spirit - play as a ghost that possesses his enemies 1 at a time, and uses their abilities against the others. Also has a bit of an adventure element where you're looking for keys, and the room to use them in, or you get the bad ending.
If you like this: http://www.romhacking.net/translations/853/
Risky Challenge is it's predecessor.
The Outfoxies - basically a 2D arena fighter
If you like Bubble Bobble:
Diet Go Go
Joe & Mac Returns
Kicker(screen might scroll a bit, but it's essentially the same idea of clearing the screen of enemies)
Nightmare in the Dark
Snow Bros. 2
Pong/Air Hockey types as characters with powers:
Battle Flip Shot
Slap Shot(Japanese exclusive)
Windjammers (also Neo-Geo)
Sort of an isometric dungeon crawler series in Dark Seal and Wizard Fire.
Elevator Action Returns(also on the Saturn) is probably the most improved sequel I've ever played.
The Gunforce games(the 2nd one separates the words in it's title) are predecessors to the Metal Slug series.
Kamikaze Cabbie - seems like an early inspiration for Crazy Taxi, you can destroy the other cars your size, but anything larger destroys you, and you need to watch your gas level.
Legend - pay enemies to be your fighting force
Legend of Makai - unfortunately I haven't spent much time with this one, but it's kind of like an arcade version of Wizard and Warriors, with a different style(you're not playing as an armored knight for example).
Seems somewhat obscured by time now, but I'd have to recommend Rally X/New Rally X to fans of Pac-Man. It's another Namco game, and you can almost see Pac-Man having taken inspiration from it(despite you and your enemies being cars). Initially this was the game that was impressing people before both released in 1980, but we know how that turned out...
Rollergames - almost breaking my initial rule, but it's a roller derby that makes a sport out of seeing which team leader can beat up the most of the other side's team.
I'll leave it there for now... I don't even play some of those half as much as I should.