I'm with the Genesis on this one. For starters, adventure and RPG's aren't really my thing.
Say what you will about RPG's, but the Genesis still seems to have more sandbox games. Pirates! Gold, Starflight, even King's Bounty, a Heroes of Might and Magic style strategy game, is fairly sanbox-like in the way that it's played.
SNES has more beat 'em ups, but it is a Japanese genre, and Japanese development definitely favored anything but the Genesis. Still, none really match up to Streets of Rage 2. There's just something about that game on the whole, and beating down 5-6 enemies at once leaves a much more satisfying feeling than 3.
The speed of the Sonic series and Desert Demolition's post 1st stage Road Runner mode(instead of default Wile E. mode) is a lot more fun than they probably should be. Especially since you need to get to know the stages before you start blazing past them to better your own time.
For all the talk of the TG-16's shooters, at least the Genesis' came out to English speaking regions, and didn't need the disc add-on to get some of the consoles very best. Can't help but notice that particular aspect tends to get overlooked. Especially when people get on a European's case about it: http://www.youtube.com/watch?v=wvlRCks1Kn8
Once you're past the shoot 'em ups, there's the run 'n guns. Hard Corps., Gunstar Heroes, Batman & Robin, I mentioned Japan/Euro exclusive Alien Soldier earlier, Ranger X, a lot of people seem to like Dinosaurs for Hire, Mercs, Gain Ground, Crack Down, etc.
Then you get the really unique stuff, like Bonanza Bros.(a game where you play as thieves back in '84/'91), Ecco, Road Rash/Skitchin', Mutant League, Devilish, Herzog Zwei(essentially the beginning of the RTS genre), Bubba 'n Stix(half platformer, half graphic adventure), Haunting starring Polterguy, The Ooze, Wiz 'n Liz, etc.
Hell, having brought up Bubba 'n Stix, you actually do get a couple graphic adventures out of Scooby Doo and Beavis and Butthead, the former even feeling like something form Lucas Arts.
I could just go on, and I've rarely brought up any of the well known stuff yet.
Even if you don't get it, there's a million reasons this console is just as loved around the world as the SNES. Even if the nerds that make up the internet's classic gaming community don't reflect that.