Well, a wizard is a spellcaster, but a spellcaster is not necessarily a wizard, if that makes sense.
Anyone who is able to cast spells is by definition a spellcaster. That's simply a description. A wizard is a human spellcaster that is also powerful enough to be a member of the White Council. He doesn't need to be a member, but he needs to be powerful enough to count as one. Elaine, for example is powerful enough to count as a wizard, but she flunked her test, so she is not a white council member. Then again, the Queens of fairy are powerful enough, but they are not human, so they are powerful spellcasters, but not wizards.
Plus, "wizard" is also a mechanical term in the game, meaning "having the wizard template", which you also don't have to have, since there are other templates that allow for spellcasting abilities.