People who exercise with their mouths closed aren't working very hard. You can't get enough air through your nose to meet your needs for oxygen when you exercise vigorously. The cross sectional area of the openings in your nose is less than one tenth the opening in the back of your mouth. That space is so narrow that when you pick up the pace, you could turn blue.
You don't need to breath through your nose when you exercise in very cold weather. Your nose warms the air much more than your mouth does, but exercise causes your body to produce such large amounts of heat that air taken through your mouth at 40 degrees below zero Fahrenheit during exercise will be warmed almost 100 degrees before it reaches your lungs. Breathing air that cold hurts so much that you lose interest in exercising and seek shelter.
Your nose clears pollutants far more efficiently than your mouth does, but people with healthy lungs can exercise safely on polluted days. Pollutants that you breathe in through your mouth can be quickly cleared from your lungs. Your air tubes are lined with small hairs, called cilia, that sweep pollutants towards your mouth where you swallow them with your saliva and they pass from your body.