A turbocharger uses exhaust gasses to spin a turbine to create boost, whereas as supercharger runs off of the drive belt. A turbocharger will make more boost because it can spin faster and does not run off of the engine, but it requires a certain amount of exhaust gas to spin the turbine at a speed that will create boost. Thus, turbo lag. In other words, a turbocharger will only produce boost above a certain RPM. A supercharger, however, will produce boost throughout the power band, but will produce less boost than a turbocharger because it spins slower and is itself a drain on the engine because it is powered by the engine.
If we want to get technical, a turbocharger is a supercharger because by definition a supercharger is ANY device that creates boost... Turbos and Superchargers both create boost, they just do it differently.