They are very different.
嘘つき is simply "liar", a noun formed from the masu-stem, which can also be used to blame someone in front of you.
嘘つけ is not a noun, but a contraction of
嘘をつけ, where つけ is the imperative form of the verb つく. This appears to mean "Tell some lie", but actually
嘘つけ! is as an idiomatic set expression that means "You're lying!" or "You must be kidding!" If I understand correctly, this is in the same vein as "Move, and you're a dead man" always effectively means "Don't move" in English. You can also compare this to how "Come on!" in English is used to both encourage and discourage something.
Since this is not a noun, you cannot say ×嘘つけになるな nor ×お前は嘘つけだ.
嘘つけ! are similar, but the latter can sound less serious depending on the context and your facial expression, whereas the former is almost always serious and accusatory. If you say
嘘つけ! to your friend with a laugh, it sounds just like English "(Ha ha,) come on!" that is said after someone said something funny.