They both do essentially the same except a supercharger is directly driven by a belt from the engine and a turbocharger is driven by the exhaust gasses.
A turbo is "spun" by the escaping exhast gasses and sometimes needs to spin up to speed. This can lead to a power surge and turbo lag. Also Turbos tend to run very hot and can glow red in some circumstances.
A supercharger is there from lower revs and gives a more even spread of torque without the surge. The downside of a supercharger is it actually uses some of the power the engine produces.