"I am no psychic" means that you are not a psychic, with 'psychic' being a person, a noun. "I am not psychic" means that you are not psychic - the adjective. The first one is referring to psychic as a noun and the second one refers to psychic as an adjective.

I agree with Catherine, and must respectfully add that "I am no" is a common construction in English, and is used rather often. However, since "I am no" must be followed by a noun, you cannot always have the same construction for "I am no" and "I am not" - in the example given, it just so happens that "psychic" can both be a noun and an adjective. If the word is only a noun, then to use "I am *not*" (rather than "I am no"), you must add an article. For example, "doctor" is only a noun (and not an adjective), so you must write: I am no doctor. But: I am not a doctor. Hope this was helpful!

I agree with Catherine. But I never use "I am no..." in a sentence. And I've never herd it been used before. Maybe in movies. I would replace it with "I am not a\an..." like in "I am not a psychic". And yeah, the first one refers to a noun and the second one refers to an adjective.