Here, check this out. This is not exactly the set-up on the MINI, but it shows you how the Eaton Twisted Vane Roots-Style Supercharger works.
The blow-off valve works on a turbo, because the exhaust gasses blow by one turbine, which in turn spins another. So the exhaust blowing out of the engine, in turn compresses the inlet air.
When you down-shift, the back-pressure between the intake manifold and the turbo, causes resistance, and the turbo slows down.
The brilliant little blow-off valve opens at a pre-determined pressure (say, near max RPM) and lets the boosted air simply escape. But the turbo is still spinning very fast. So when you slam the shifter into the next gear and hit the gas again, the turbo is already up to speed.
A supercharger is physically attached to the engine, so it cannot remain spinning quickly when the engine slows down. So a blow-off valve would be pointless.
Now, even with a supercharger, there are some circumstances when the engine is spinning quickly, and the intake is not drawing much air. Like when engine braking on a downshift. You could simply 'blow-off' this air, but since the Supercharger is a drag on the engine, the by-pass valve opens and dumps the unused charge back into the supercharger's inlet. Since the inlet pressure and the outles pressure are similar, this reduces the drag the supercharger places on the engine. This is preferable to a blow-off or dump valve.
At least that is my understanding of it.
