As adjectives the difference between psychic and cognitive
is that
psychic is relating to the abilities of a psychic while
cognitive is relating to the part of mental functions that deals with logic, as opposed to affective which deals with emotions.
As a noun psychic
is a person who possesses, or appears to possess, extra-sensory abilities such as precognition, clairvoyance and telepathy, or who appears to be susceptible to paranormal or supernatural influence.