Are precognition and premonition the same thing?  Do some experience this more than others?

Their outcome is essentially the same, but "precognition" has a connotation of being internal (i.e., coming from within, internal forethought) and "premonition" has a connotation of being external (i.e., coming from without, external foretelling).

As for psychology or parapsychology, they are parapsychology. Ever since the paradigm shift to behaviorism, psychology has gone from an art to a science, a hard science that strictly applies the scientific method. In scientific psychological studies, the behaviors of precognition and premonition haven't been proven. Incidence of seeming precognition and premonition have always fallen within the margin of error of standard probabilities, of luck. So sometimes someone claiming such abilities does correctly predict what a drawn card or a roll of the dice will be, but they correctly predict it no more often than someone who doesn't claim such abilities and no more often than one would expect based on the laws of probability.