Superchargers come in various types – positive displacement, centrifugal, twin screw, etc. Positive displacement gives instant power (no waiting), but bonnet clearance can be a problem. Twin screw also gives instant power but can create high intake temps.
Centrifugal superchargers make it easy to intercool, therefore keeping intake temps low. They are not as instant in making boost power but make more with RPM – lineal power that makes it nice to drive at cruise RPM, and they just get stronger and stronger at the top end.
Any supercharger uses up power to turn being belt driven. Turbochargers make boost from exhaust pressure, therefore not using engine power to turn a belt. Huge max power can be made but usually with the disadvantage of turbo lag, meaning the bigger the turbo, the more exhaust pressure needed to spool up, which is where the lag comes from.