The Wastegate, similar to a "pop-off valve", purges unused air charge from the engine.
Basically, think of the Turbo or Super charger's blast of air as a "charge". When you lift off the throttle, there is still a charge on it's way to the engine. If that charge enters the engine without the fuel (because you lifted) it causes an ill-mixture and detonation can occur.
The wastegate simply diverts that charge out of the system when you lift between shifts or on deceleration, saving both the engine AND your pocketbook 
The effect becomes worse with higher boost levels. A little is bad enough, but if you are running 9 or more pounds of boost, you're going to grenade your engine eventually.
Anyone telling you that you can run a high power Turbo WITHOUT a pop-off valve or wastegate is setting you up for costly repairs.
As for the difference and preference between the two, I have a 14.9 lb boost Turbo on my Lotus Esprit and a 6 lb boost Super Charger on my Durango R/T...
The Lotus "kicks" at a pretty low RPM, so Turbo lag is almost non-existant. As long as you get your RPMs up to around 2500 before you let the clutch go, you are starting out with boost. Of course, as the RPMS come up, so does the boost. So at 4500 the car acts like you've been shot out of a cannon!
The Supercharger on the Durango, on the other hand, "kicks" very early... just over idle in fact. The pull is aggressive but the maximum horsepower is achieved at around 4800 RPM.
As with both chargers, impeller size and design will change the power curve so you can get Turbo chargers that will deliver power sooner ans well as Super chargers that will deliver power over a longer RPM range. It's all up to you... and your pocketbook 
DKOV -