They both do the same thing (i.e force the air/fuel mix into the engine at great pressure), the difference is in the way they are driven. A turbocharger is driven by re-routing the engine's exhaust gases into the turbo unit to turn the impeller. The upside is that it's "free" power, so to speak. The downside is that at low revs, there's isn't enough pressure in the exhaust gases right awayto drive the turbo. That's why turbo engines are a bit "soft" at low revs, any why you get lag. Once the revs rise & the exhaust pressure being fed into the turbo reaches the critical point, you get this sudden burst of power.
A super charger does the same job, but is driven directly off the crankshaft, so is permanently on. The upside is that there's no lag. The extra power is always there, and the amount of forced induction rises in direct proportion to the engine speed, so the power delivery is much more linear. The down side is that the always-on charger uses more fuel.