What are the operational differences between an airplane equipped with a constant speed propeller and one with a fixed pitch propeller?
A constant speed propeller runs at a specific rpm by changing the blade pitch. When the blades are at a lower pitch (i.e., flatter angle), there is less air resistance, so the propeller puts less drag on the engine and runs faster. Additional pitch causes more drag and slower RPM. A governor (or something similar) will adjust the blade pitch to keep the propeller running at a particular rpm, even when more or less power is produced by the engine.

A constant speed prop is typically found on more powerful engines. Constant speed props are more efficient, but are more complex and heavier than fixed pitch props.

Some planes have a prop speed control for the pilot to adjust the propeller rpm, and some, such has the Cirrus SR22 and Pilatus PC12 have a single prop speed for flight.

The operation of a variable pitch prop depends on the plane. For example, in a Cirrus SR22 you don't even have a propeller pitch control. You operate it just like a fixed pitch prop. With a Pilatus PC12, the only prop pitch adjustments are when you feather the prop on engine shutdown or reverse the prop pitch for landing or ground operations. In a Beech Bonanza A36, you adjust the prop pitch for max RPM (minimum pitch) on takeoff, then add pitch for cruise flight, and max RPM for landing (in case of go-around).