I think the various posts in this thread collectively address the question pretty well as far as running goes. My observation/experience is that most runners breathe through the nose and mouth, simultaneously, in various proportions according to their personal physiology, and the oxygen demands of the race or workout.
But to address an interesting aspect of the original question, why the many general recommendations to "breathe through your nose"? This is because that is the correct advice for normal, sedentary, activity. Breathing through the nose provides filtering, germicidal, and warming functions not provided by breathing through the mouth. In normal or sedentary activity, it is much healthier to breathe through the nose. Habitual mouth breathers suffer many health and respiratory problems which do not affect nose breathers.
The mouth provides an additional breathing channel for use during high-oxygen demand activities, such as running and other intense exercise. Also singing, and to some extent, talking.
Breathing through the nose does, to some extent, stimulate the brain, but not by providing oxygen directly to the brain, but by the generally stimulating effect of the physical passage of air over the nerves in the cranial region of the upper nasal passages. It's easy to feel this stimulating effect directly by concentrating on the feeling of the air passing through the nasal passages.