They may only be words but in the world of fantasy literature are they synonymous? Can they be used interchangeably? Perhaps. And they aren’t the only words used to indicate someone who can use some version of magic. What about Prestidigitator, Conjurer, Magician, Enchanter, Warlock, Sorcerer, Illusionist, or Necromancer? Maybe in any particular fantasy world, each of these terms might connotate a specific status or position within the magical hierarchy. In my fantasy series, The Tirumfall Trilogy, mage, wizard, sorcerer are all the same though there are varying ranks of them designated by other terms and titles. What about your favorite fantasy world? Or if you’re a fantasy author, how about in your world?
You can see what I do with it here: https://www.amazon.com/gp/product/B071J8LRJP/