One of the biggest problem with a pure mage is that you'll always be mana starving the first few levels. The beginning spells are also pretty garbage; they deal pathetic amounts of damage, they eat through mana like crazy and you have to be close to enemies to even start to hurt them. All those three combined equal to a very dead mage.
Once you start getting some actual decent spells (by decent I mean spells that you can safely cast at a distance), then you can bring some pain to the enemy.
I had a playthrough going on with a pure mage, and every single level up I would increase Magicka, because Health is for chumps. Of course, that also meant that I could be one-shotted by pretty much anything in the game. It was fun having to figure out a way to kill everything without eating an arrow or being impaled by a sword. Or, you know, just abusing the hell out of the Impact trait and stun-locking enemies. But here's the thing about them: while at around levels 20 to 30 you feel like a badass, killing anything that stood in front of you, above that and you'll see that it's going to take a long, long time to kill anything. Spells don't have a great scaling in terms of damage, and if you're playing vanilla, then good luck trying to kill a dragon, or anything, really. Further, you don't have a great deal of spell selection, and you'll quickly start relying way too much on summons/followers to tank AND kill enemies, and lastly, the Master level Destruction spells just ain't that good because of how long the animation is. That, and the damage is negligible at higher levels.