You can wear out a normal HDD just by operating it on an uneven (not flat) surface. Inside are 1-3 platters with a needle (think similar to a record player, but smaller and spinning much faster). Large enough bumps while the disk is in use will cause the needle to scratch the platter. Just as a scratched record ruins a song or more a scratched platter at the very least will prevent the system reading files in those locations (and remember files are often fragmented across the entire disk).

If the school laptop is off when the dropped the needle is in a safe location away from the platters so the risk of HDD damage is less in that instance (could still cause damage in other ways though). A crude comparison to comparing HDD shock resistance might be a watch/timepiece. The genuine expensive articles are going to generally last longer than the cheap knock-offs.

An SSD has no moving parts so any old one will do the job as far as shock resistance goes, however you should research reliability of certain brands/models as some are better than others. I wouldn't touch an early-generation model.