They do the same thing: They use some sort of fan to drive more air into the engine. The engine works like it is bigger than it really is.
The turbo does this by using the power from the exhaust gas. The super charger gets its power from the crank case.
The turbo runs hotter but gets "free" energy that would have been wasted. But in reality it doesn't take that much energy to drive the turbo. I think the supercharger is "better" in that it runs cooler, but that's a matter of opinion.