As far as I know, breathing through the nose might be the healthier option if you don't need the extra oxygen (for say, intense uphill efforts). Breathing through the nose will optimize the temperature, humidity of the air you breath in and also makes any pollution reach your lungs a bit less probable.
There are also some other subtle effects, but since I'm no expert, I won't go into that. You might want to research that if you are very interested.
During an intense effort, I would say deep (diaphragm) breathing through the nose and the mouth is the most effective way to fill your lungs with fresh air. You might want to practice diaphragm breathing off the bike as well to get used to it. It isn't very natural for the first time.
So the tldr version is:
- Nose when not intense effort
- Nose and mouth when intense effort
Also try to keep your mouth humid when you breath through it (drink very small sips of water frequently). A dry mouth is very unpleasant and also unhealthy (gums, throat).
On the cotton wool (it's pure speculation): maybe something is put on them, probably to clean up the upper respiratory tract from mucus.