When flying at higher altitudes, does it matter if you have a turboprop vs a piston engine driving a propeller?
In general, a turbine engine will produce more power than a piston engine at higher altitudes. A turbocharged piston engine, with compressed air intake, will lose less power at altitude than an normally aspirated piston engine. A turbine engine will lose even less.
As a rough example, a normally aspirated Cirrus SR-22 has a service ceiling of 17,500 ft. A turbocharged SR-22 has a ceiling of 25,000.
A turbocharged piston Piper Malibu Mirage has a service ceiling of 25,000 ft., while the turbine powered Malibu Meridian has a ceiling of 30,000 ft.
The turboprop Pilatus PC12 and Socata TBM-850 have service ceilings of 30,000 ft. or above.
Of course it does!
One simple way of looking at this is that piston-powered (well cylinders) are constant-volume engines (ie. 1200cc). As you go higher the air becomes less dense so there is less air mass (and fuel) getting into the engines. If the density becomes half what it was at sea level, you now have in effect a 600cc equivalent engine, etc. Power will be less and less.
A turboprop is a constant pressure engine, so as you climb, you kinda need to go faster and faster, but basically you still burn the same amount of air/fuel. You will lose power as you climb in a turboprop, but that's mostly cause the propeller loses efficiency at the higher speeds you fly higher up. The turbine itself is still as efficient as on the ground.