Nose breathing is much better for you
Functions if the nose:
Filtration of unwanted particles
Warming of air before it reaches the lungs
Humidification of air so it doesn't dry the lungs.
Also there is somethig called a naso-pulmonary reflex which means that when you breathe through the nose the little air sacs in your lungs (the alveoli) dilate and exchange gas better - so yes, you do get a deeper breath. This reflex is thought to be mediated by Nitric oxide produced in the paranasal sinuses.