As I understand it, a supercharger is powered by a belt driven off the engine, so you get the cold air blown in right from when you set off, but some of the extra power is lost to keeping it going. With a turbo charger you don't get the benefit until you've reached a high enough speed so that exhaust air going out is powerful enough to spin the turbine in your turbo.
You can get hyrbid turbos where a smaller, less powerful turbo works until the larger one can take over. Or I'm told Nitrous Oxide can make up the extra cold air burst until you're going fast enough to just use the turbo.
All I really know is the turbo whine sounds great as you're pelting along, and so does the dump valve hiss when you change gear.