Storage is the filing cabinet: it holds documents, photos and programs while the machine is off. Memory is the desk: it holds only what the computer is actively working on right now, and it empties every time you shut down.
Running out of each feels different. A full drive stops you saving files and blocks updates. Full memory makes everything sluggish, because the computer starts shuffling working data back and forth to the much slower drive to cope.
That shuffling explains why a machine with plenty of free disk space can still crawl with twenty browser tabs open. The tabs are competing for desk space, not filing space, and freeing storage will not help at all.
Knowing which one is short tells you what to change. Persistent slowness with many programs open usually means more memory or fewer open programs; warnings about saving files mean the drive needs clearing or replacing.