The turboprop engine produces less power at altitude, because the mass stream through the engine is smaller: there is less air. The turbocharged piston engine also produces less power at altitude due to less air entering the cylinder. Both have a compressor, and some effects of altitude on power generation are indeed similar, but the design conditions differ due to the great differentiator: weight.
A turboprop engine's weight increases much less as a function of max power at sea level, than that of a piston engine. The turboprop can be designed for cruise conditions, and take the surplus of power at sea level as either:
- a bonus, so that much more power is available at sea level and power output decreases gradually with altitude;
- a liability for the gearbox, so that power limitations at lower altitude are applied by the FADEC.
If the same approach was followed for a piston engine, the weight penalty would be much higher than for the turbine engine. So piston engines are designed for maximum power at sea level, and this power is maintained as long as possible with increasing altitude, by pumping in more air through the super/turbocharger.
Note that the turboprop with derated power and the turbocharged piston have very similar characteristics, both deliver a constant power as altitude increases, up to the critical air pressure...
The weight advantage of turboprops over piston engines was demonstrated in the 1960s by the conversion of the Cessna Skymaster into the Conroy Stolifter: the two Skymaster piston engines produced 155 hp less and weighed 117 kg more than the single TPE-311 turbine engine that replaced them.