Well, we have "tap water", and I've heard people refer to a "tap" as the source of water (as in "get some water from the tap"), but I've never heard anyone call a faucet a tap. A tap would be the thing from which you get beer out of the keg.

We also have spigots, but that's usually an exterior-wall-mounted faucet. Occasionally you hear "sillcock" as another name for a spigot. I believe that it's the more technically correct term.