'False' is more common in British English. But there is a difference. 'False' simply means not real. 'Fake' means that you are trying to trick or deceive someone. So whereas I might refer to someone having 'fake ID' or a 'fake Rolex watch, I wouldn't use 'fake' to refer to grandma's plastic teeth - I'd call them 'false teeth'. 'False' can also be used in many situations where 'fake' can't. A false assumption, or false logic, for example. These are wrong, but you're not trying to trick anyone. If you remember than 'fake' is something designed to deceive, then you can't go wrong.

I think in almost any case in American English we would use the word "fake" to describe artificial things like that. We do call a counterfeit/forged ID a "fake ID." False generally means "not true." Fake generally means "not real." Some people call dentures "false teeth," but that's the only example I could think of for false. Finally, the singer of the song you're referring to is probably placing emphasis on the word "fake," to attach its meaning—not real—to the person being addressed. It sounds like an accusation of disingenuousness in this context. But that's artistic interpretation :)

Hi Shoji, Yes the words are interchangeable. They mean the same thing. Hope this helps.:)