I agree with Mariana about the person/object distinction. A thing can be phony - google 'phony diamonds', for example, and you'll get plenty of hits - but you don't often use the adjective 'fake' for a person. 'Phony' for a person often has the meaning of insincere and unconvincing. If someone is behaving in a way that you can tell is insincere - whether they're a salesman trying to sell you something or a celebrity trying hard to appear modest - you'd describe them as 'phony'. Fake eyelashes, fake suntan and a phony manner, for example. Another difference is the way the adjectives work. 'Fake', as an adjective, is mainly used attributively, before the noun it describes - fake passport, fake Rolex, for example. 'Phony' can be both an attributive or predicative adjective, i.e. before or after a noun. For example, you might say 'His upper-class accent sounds phony', but you wouldn't often use 'fake' in this way. Both words can be used as nouns. If you describe someone as 'a fake', they are deliberately lying you about who they are and what qualifications and abilities they have. We've had quite a few 'fakes' on this site - people who've claimed that they are native speakers/professional teachers of a language which they barely know. A 'phony' isn't necessarily lying about who they are in the way that a fake is. A 'phony' is more like a politician who has a carefully cultivated persona of being sincere and caring when everyone knows perfectly well that it's a facade.

They work as synonyms. However, I would use "phony" to people and "fake" to objects. This is my personal use, not a rule.