What makes a constant speed prop different from a fixed pitch prop, and what are the operational differences between airplanes with the two?
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).
The main operational difference is in power management. With a fixed pitch propeller, if you want more power for climb or speed, you add throttle. When you pitch up, the the RPM will slow due to less air moving through the propeller: suddenly instead of inertia helping you along, your engine is having to work to keep you going. When you pitch down, the RPM will increase due to more air moving through the prop. In this case you have to reduce throttle or you will overspeed the engine.
With a constant speed propeller, engine RPM is largely unaffected by the attitude and airspeed of the aircraft. The problem is that if you're flying along at cruise speed with your propeller in cruise configuration and decide to pitch up to climb, your propeller is not able to impart the full energy of the engine into the air. Your climb performance will be diminished or non-existant until you reset the prop to a low pitch. So the main thing to consider is that with a CS prop, maximum power is not always instantly available.
You can also damage the engine easier with a CS prop, in my opinion. With a fixed pitch prop if you go into a dive, your RPM will increase giving you visual and auditory clues that it's happening. With a CS prop, you can enter into a dive and possibly go all the way to Vne without the engine noticing a thing. The problem arises when you're going fast and decide you want to push the prop lever forward: your engine suddenly jumps hundreds or thousands of RPMs in an instant. This is extremely bad for it and that's why many aircraft manuals say not to increase to max RPM above a certain airspeed.
Finally, although it is more than made up for in efficiency, engine management is just more complicated with a CS prop. Say you're wanting to maintain a stabilized approach at 100kts. You need to find some combination of throttle and prop settings that gives you enough power to maintain a stable sinkrate, and either has enough drag to slow you down to 100kts, or has minimal drag, allowing you to keep your speed up.