Nose is meant for breathing and it has filters to prevent dust and other particles entering the lungs. Therefore obviously nose is better and when our body breathes naturally without our attention we breathe only through our nose. Unless we are suffering from a blockage in the nose due to sinus or cold or some viral infection.
However, for certain types of breathing such as Conscious Breathing as I teach, the exhale breath is done through the mouth in a long, slow exhale while making a loud sigh sound. The in breath is allowed to happen effortlessly through the nose. Only exhale needs effort to ensure that the lungs get completely emptied before pausing and allowing the in breath to happen naturally. This is a deeply relaxing breathing process and it also helps instantaneously quieten the mind.
This practice is to be done only for a few minutes and once we feel the shift in our state of mind from restlessness to a peaceful or still state, we can revert to normal breathing through the nose with both the in and out happening effortlessly while just holding the breath in our awareness. Practice this right now and experience it first hand.
To learn more about breathing and other deeply healing practices please visit my website or read my book titled Inner Alchemy.