Wicca is one form of paganism (just like Catholic is one form of Christianity), focusing more on the 3 faces (or phases) of the goddess (the maiden, the mother and the crone), which is pretty much a trinity to the life cycle of youth, adult, elder. Those of us who are pagan, but not Wiccan, just basically choose different names to put to the gods, as most of the ancient religions (Roman, Greek, Celtic, Eqyptian) had many of the same ideas with different names. My choice is the Egyptian gods, Earl chooses the Celtic gods.
Magic is one aspect of any pagan faith. As Sue said, it gives the person a more active role in controlling the environment around you. The "rule" that I've always seen with magic or "spells" is that it must not ask for personal gain, but for the good of the situation or the best possible solution. (I.e. you couldn't/shouldn't ask to win the lottery, but ask for the financial situation to become manageable.)
The pagan faiths really, IMO, put one closer to nature and appreciate the complex world around us. (Not that other faiths cannot do the same thing, but it is more of the focus of the pagan faiths.)