I've been very interested on this topic some years ago and I read quite a lot of studies.
All together, there's no straight answer to favor one, because each is preferred in certain circumstances.
Let's start with the beginning.
Nose breathing is favored in the warm up, because it delivers carbon dioxide faster to the body.
Also, it helps the nostrils to dilate faster.
But later nose breathing can become a problem, because it lowers the expel rate of carbon dioxide, leading to reaching the anaerobic state earlier than ideal.
Best use of breathing, in my opinion, is this:
- warm up: nose only
- early part of the run: inhale through nose, exhale through mouth
- last part of the run: inhale through nose, but don't block the inhale through mouth. If possible inhale with both. Exhale through mouth.