All these words are similar, in that they describe a device which controls the flow of a liquid through some sort of pipe. Technically, they are interchangeable, but it practice there is a slight difference. "Tap" and "faucet" would be used if you are referring to the sink in your kitchen. They are interchangeable. They would usually be used for something indoors that controls the flow of something you drink. "Tap" is also used to control the flow of something from a cask (barrel) - usually beer. In a bar, if you ask for draft beer, it will not be taken from a bottle but from a tap, which is connected to a barrel of beer. "Spigot" and "stopcock" are usually used for outdoor pipes. A "stopcock" usually is used to describe something more basic and primative. "Spigot" is also used figuratively to describe something that controls the flow of money. All these words mean pretty much the same thing. The exact usage may be different, depending on the pipe or flow system we are talking about.

In the US the most common of these is a faucet, the fixture on a sink. 'Spigot' has mostly fallen into disuse, though people know what it means. A stopcock is usually a valve in a water line, so that's a more technical term. I do hear people use 'tap' to mean a water faucet: 'turn on the tap'. I think this might be a regional thing, but I'm not certain.