Do turboprop and piston engines perform differently at higher altitudes?
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.