Ok, I've been doing a lot of studying on turbos lately, so I apologize if I ramble on. If I need to clarify, please let me know. Also, most of my knowledge resides with turbos, however I will describe superchargers as best I can.
Turbos and superchargers by function do the same thing; force induct (push) additional air into the engine. How they do it is where they differ. Your use of the car will define which one will better suite you.
Superchargers are belt-driven and are functioning all the time. Power is available immediately upon acceleration, even from a stand still...which is why they are ideal for dragsters. However, their power gain is limited by the size of the supercharger and the RPM of the engine. Also, because there are many moving parts, they are less reliable and there are more things that can go wrong.
Turbochargers, on the other hand, can be more reliable (if preoperly treated) because there is only one moving part; the turbine powered by exhaust gases. The ever present word 'lag' is basically the time it takes from the time you begin WOT (wide-open throttle) to the time the turbo kicks in (or spools up enough to increase the intake pressure). The larger the turbo, the greater the lag. Lag is not necessarily a bad thing. If you think about it, a car with no turbo is experiencing full-time lag. Smaller turbos have less lag, but then again they deliver a smaller boost in power. If you do most of your driving in the upper RPM band of your engine, lag isn't really much of an issue because there is plenty of exhaust gases flowing to keep the turbo spooled up.
Also, the larger the turbo, the greater power gain you will receive from it. Also, the larger the turbo, the less restrictive of an airflow your car will have. The draw-back is that you will have to wait a little bit longer to get it.
Turbos are more efficient than superchargers and the power potential is far greater than that of a supercharger. They net a higher gain per cubic inch than superchargers do.