A hard drive stores data on spinning platters read by a moving head; a solid-state drive stores it in flash memory with no moving parts. That difference explains almost every practical distinction between them.
Speed is the obvious one. Solid-state drives respond almost instantly, which is why replacing a hard drive is the single biggest upgrade an older laptop can receive.
Failure modes differ more importantly. Hard drives often warn you — clicking, slow response, growing bad sectors — while solid-state drives tend to work perfectly until they abruptly do not.
That silence is why backups matter more on modern machines, not less. Recovery from a failed flash drive is expensive and frequently impossible.