My understanding:
A Turbo- charger uses exhaust gases to spin a compressor on the intake air and feed it at a higher pressure to the cylinders, providing more oxygen to the fuel for added combustion.
a Super- charger does the same act but is driven directly by the engine shaft.
Turbos add to the efficiency of an engine by using the hot waste gas, but suffer from the need for cooling to last long, and can only provide power e.g. "spool up" once the engine is running, and this is "lag"
Superchargers "steal" some shaft horsepower but in the end provide more direct constant compression.