A difference was the two stage supercharger in the F4F was good for 1000hp at 19,000ft. The same P & W R-1830 when given a turbocharger (and using the same engine driven supercharger) was good for 1200hp at 25,000ft in the last versions of the P-43. It did require a bit more weight and bulk and the turbo (and controller) wasn't really ready for front line use.
The problem with single stage single speed superchargers is that they can only be optimised for one altitude. The Best engine I know of for illustrating this is the Merlin. The Merlin III used in the Hurricane I Spitfire I and a several other British aircraft would hold 6lbs of boost (around 42in ) to 16,250 ft with the engine turning 3000rpm which spun the impeller at 25,764rpm.
Please note that anytime the engine was doing 3000 rpm the impeller was spinning at 25,764rpm regardless of altitude. The air at 16,000ft is at 16.21 inches of Hg so the Merlin supercharger was compressing almost 2.6 times. If the throttle had been opened all the way at sea level the supercharger might have been able to produce 77in of manifold pressure which would have certainly either caused massive detonation with the fuel available or destroyed the engine even without detonation. By restricting the amount (volume of air) coming into the supercharger at sea level the engine made 880hp for take off at the same 3000 rpm and same 6lbs of boost (some books give 6 1/4lbs).
There are two sources of heat rise in a supercharger. One is the simple heat rise of the work of compressing the air/intake charge even assuming the compressor is 100% efficient.
The other is the fact that most pre-WW II and WW II superchargers were not very efficient. If 70% of the power applied to the input shaft of the supercharger actually went to compressing the air it was a very good supercharger indeed in early WW II. The other 30% of the power didn't just disappear. All but a tiny percentage (under 1% for friction) went into heating the air over and above the heating done in actually compressing the air. You wound up with some
very hot air going into the engine.
A supercharger that was 65% efficient saw 35% of the power applied to the input shaft turned into excess heat of the intake air/charge and so on for lower efficiencies.
the power needed to turn the supercharger impeller was proportional to the speed of the impeller (or more properly to the speed of the tips but if we are talking about the same supercharger the impeller speed will do). Obviously reducing the impeller speed will require much less power and also much less power going into heating the intake charge.
The supercharger on the Merlin VIII engine was the same as the Merlin III except it used a set of 6.313 gears instead of 8.588 gears the overall change was an increase in take-off power 50 1080hp at 3000rpm using 5 3/4lb boost. Less power used to turn the supercharger, throttle plate open further reducing pumping loss and a much cooler/denser intake charge. trade of was engine topped out at 1275hp at 3000rpm at sea level suing 9lbs of boost.
The heat of compressing the air, especially when factoring in the waste heat, rose faster than the outside air temperature dropped with rise in altitude.
The heat rise of a Merlin XX engine with Hooker supercharger (more efficient than earlier models) was figured to be 148 degrees centigrade at 3000rpm in high gear almost regardless of altitude. at 35,000ft the intake charge temperature in the intake manifold wass about 95 degrees centigrade. In part due to the fact that much less work was being done. Only 84.5lbs of air and fuel per minute going through the supercharger ( and outlet pressure of 26.56 in) compared to the 144lbs at 20,000ft and the output pressure of 48.24in.
I hope this isn't too confusing. But superchargers, despite only having one moving part are not always simple to figure out