The biggest difference is usually in the implementation. You've got to choose the right hardware - size of the blower/compressor - and mount it right. Get the right turbo and it'll have low end torque. Get the right supercharger and it'll keep pulling hard to redline.
It's actually harder to do a good supercharger system in a lot of ways, as you've got to get your belt alignment juuuust right and manage to transfer a fair bit of torque through the belt without resorting to massive tension. An MP62 spinning fast enough to make 10 psi of boost on a Miata eats up 36 hp - and that's all run through the belt. Then you have to deal with the driveability problems due to the big throttled volume, and now we get into intercooling. It's a lot easier to put together (or buy) a turbo manifold and downpipe.
However, superchargers are a bit easier to tune off-idle because they're quite linear. x rpm * y throttle = z boost, every time. They're also more resistant to detonation thanks to the low backpressure, which does help make up for the lack of intercooling somewhat.
Generally speaking, I like turbos better. They're just better mannered, with less vibration and noise when you're not on it. The shriek and instant hit of a super is a lot of fun, though. There's a hard edge to a supercharged car that you just don't get on turbos unless you're romping around at high rpm.
Here's a comparison you might enjoy. The Janel in the story is my wife. Yes, it's being used to sell FM turbos. But she's convinced, and don't you even think about taking her turbo away from her.
http://www.flyinmiata.com/turbos/janel.php