Turbos don't take any power to turn, it's all free from the exhaust. (Dragsters excluded) A turbo will have a larger power reservoir to draw from than a blower and won't rob the motor of power due to parasitic drag on the RA. Figure - roughly - around 20% of motor's horsepower is spent turning the screw so the blower starts at a power deficit and has to make all that power back up and more. You can only get a pulley so small to up the boost. The upside is that the power is always there, there no lag that is associated with a snail, you push the 'go' pedal, the blower pulley turns in relation to the crank pulley. The turbo 'lags' because of the time it takes to build enough exhaust velocity to spin the turbine. However once that power comes on, its all addition - not counting the extra heat generated by that 1600° lump of cast iron hanging off the motor.
The problem with turbos is that they are externally more complicated. Oil feed and returns to run, mucking with the exhaust, etc... whereas a typical modern supercharger is slapped on the intake manifold, squirt some more fuel, run another belt and you're done. I'm generalizing of course, but you get the idea.
Show me a practical supercharger that put the squeeze on the incoming air to the tune of 30psi without melting into a puddle of aluminum. A turbo can do this all day long with the supporting modifications.
Cliff notes - turbos are more efficient, generate more power, but typically provide power later in the RPM band, produce more heat and require a lot more room and plumbing than a blower. A blower provides power all the way through the RPM band, is simpler to install, but is less efficient, and unable to achieve high boost levels.
I'm a recovering turbo junkie, but I give it about 6 months once my new motor is running before I start looking to boost it!