Is Ice Wizard better than Princess? It really depends on taste. I think that princess is better because of the high damage, high range and splash damage. The ice wizard can slow movement speed and attack rate but even with that, it will die quickly due to its bad range. It's damage is low and won't deal much before dying. Princess however, can shoot the tower without being in the range of the tower. It deals a nice high damage but with very low health. The ice wizard is different because of its low range and low attack but high health. This can help him live longer but if he is sent alone, he won't deal much.
Ice wizards are better than princesses because although princess mighty have low elixir cost it can be defeated easily by arrows. Not only that someone could easily turn the battle against princess and hit the tower with a lot of health. An ice wizard could slow movement and do damage.
Even though Princesses can easily be defeated by arrows, that could be used as an advantage. It would be an equal exchange of 3 elixir and you will be able to place maybe goblin barrel or minion horde when he can't counter them with arrows. When you say someone can easily turn the battle against the princess, by that time you will already have gained back the elixir you spent to place your princess. Also, you could easily turn the battle around against an ice wizard too so they are equal on that part. The ice wizard can do damage but not as much compared to a princess.