It's more than just a name. It means they didn't go to school together, and they didn't learn from the same teachers, or read the same books. It means that a PC wizard could have been a mage instead, if they had made different life choices. (It kind of reminds me of playing 4E, and wishing I could play as a mage instead of a wizard - recharge spells were way more my style than AEDU, and mages didn't have to follow the magic item treadmill in order to keep their stats up to par.)
And it also means that we have no idea what the rules for creating and advancing a mage actually are; we have a snapshot about what a typical mage looks like at a particular point along its advancement, and everything else is speculation.