In the current parlance, "cap" is an ambiguous word. If I say "I have a new cap," I could be referring to a thing I wear on my head, or to a thing that keeps the fizz in a bottle of soda, or to a limit on new spending, or...any number of things.

Whereas a hat is an article of attire worn on the head. Period. If someone tells you "Matt has a new hat," you might not know if it's a porkpie, a fedora, a mitre, or a big purple velvet pimp hat with a huge plume, but you know he's got something on his head.