SNES: Much better plot, writing, and memorable characters. Gameplay is much more RPG/Puzzle based. You'll need to think your way through most tough parts of the game, though you'll probably spend a bit of time gathering karma and nuyen too. You also have to get used to the dialog interface. You basically "talk" to people to get new dialog subjects, and then use "Ask about..." You get somewhere around 20 or 30 subjects per area so it'll take you awhile to go through all of them with everyone by the time you're done with an area, but it's fairly easy to get critical info fast.
Genesis: Almost a dungeon crawler. Good writing but not as good as the SNES. You have far and away more gameplay options by virtue of the fact that you can actually choose one of three classes to be in the beginning; Shaman, Street Samurai, and Decker. You can play as the other two classes by hiring Shadowrunners and using their skills. You'll spend most of your time doing runs for nuyen and karma so that you can take on the next story related quest. The combat is faster paced than the SNES one and generally more action oriented, but you'll win more through virtue of firepower than dodging skills more often than not.
I put a ton of hours into both, but I'm a sucker for the Shadowrun universe and cyberpunk in general. I think the Genesis version takes longer to beat than the SNES one,so you might try starting with it.