The Americans are technically more correct than we are.
When used as nouns, faucet means an exposed plumbing fitting, whereas tap means a tapering cylindrical pin or peg used to stop the vent in a cask.
So a faucet is a device with a hand-operated valve for regulating the flow of a liquid. It originates from French and appeared in Mediaeval English writings.
Technically, a tap is used when there isn't a pipe, like when you tap a keg of beer, or tap a maple tree for syrup. Tap is a little unusual because it can also be the act of tapping as well as the device you use to control the flow. A tap has the sense to an American of poking a hole in something that has liquid in it, and being able to keep the liquid from just gushing out with some sort of device. Faucets and spigots are plumbing, with connectors and pipes.
What seems to have happened is that we Brits use 'tap' for all three in a generic sense and this emerged from Old English in Tudor times onwards. Spigot resulted from French by the way.