To a pilot, does turbocharging differ from supercharging?
I have some of the answers:  

•Is it easier or harder to overboost a supercharger? If you do, are there waste gates?

This is an important point.  Waste gates, when opened, allow exhaust to bypass the turbocharger, reducing manifold pressure by slowing down the turbo.  Older turbos had manually adjusted waste gates, and even fixed waste gates.  The pilot would have to monitor boost pressure and adjust the waste gate to avoid overboosting the engine.  Newer turbos have automatic waste gates.  Superchargers, being belt or gear driven, do not have waste gates as the supercharger is not dependent on exhaust gas flow.  I do not know if they have manifold pressure releif valves in case of overpressure but I doubt they would.  

So a key difference in a pilot's operation of the engine is that in a manual waste gate engine, the manifold pressure gauge needs to be scanned frequently and the waste gate adjusted accordingly.   Less important in an automatic waste gate or supercharged engine.  

•Do engines with superchargers need a cool-down period before being shut off?

No.  The cooldown period in turbochargers is necessary to cool down the bearings in the turbocharger after high cruise power settings.  Idling on the ground allows cooler oil to flow through the bearings, removing excess heat.  Look at your exhaust gas temperature gauge - this is the temperature of the gasses flowing through your turbocharger.  The oil present in the turbocharger bearings will "coke", or turn into a solid at that temperature.  Upon landing, a slow taxi to the ramp will usually provide sufficient cool down time.  Usually!  Time this and don't forget to cool down as recommended.  

Superchargers are geared to the engine (or belted) and do not use the hot exhaust as their source of power.  So even though they can get quite hot, they don't approach the heat of the exhaust gasses.  

•Any other operational differences that would be meaningful for pilots to know.

This is not really an operational difference, but in addition to turbocharged and supercharged engines, there are also turbonormalized engines.  TN engines are turbocharged engines with their boost limited to atmospheric pressure at sea level.  Say 30 inches, not the 40 inches you might expect from a turbocharged engine.  The TN engine permits sea level performance up to a critical altitude, above which the TN engine cannot maintain sea level manifold pressure.  In other words, performance at a sea level takeoff will not be materially different from that of a naturally aspirated engine, but as you climb, the power will not decline as you climb.  So your Leadville CO takeoff will be the envy of the stock C-172 crowd.