Both idiomatic expressions can be used to mean it's clear that someone is lying, but the they have different grammatical structures and meanings.
The meaning of "plain as the nose on your face" is "obviously true". The phrase can only be used to describe statements of fact. "He's lying" is a statement of fact. This sentence means: "'He's lying' is obviously true."
The meaning of "stands out a mile" is "highly contrasting to the background", or "easy to identify/pick out from among many". It can be used of anything in any context, not just statements. The sentence means, "Compared to all the things we hear people say every day, it is easy to identify that he's lying."
For an example that contrasts the two expressions, consider someone who attends a formal party wearing pyjamas. You could say, "She sticks/stands out a mile" because she is dressed so different to everyone else at the party. However, you could not say "She is plain as the nose on your face" because "she" isn't a statement, so it makes no sense to say she's "clearly true".