I think that's an easy one. The nose is for filtering the incoming air and warming it up. Breathing through your mouth is for extreme situations when you need more oxygen at a time than you can get through all the filters.

The technique trainers recommend combines both advantages.

Breathing through the nose has a few benefits.

The air you breathe will be slightly warmed on its slightly longer journey to the lungs meaning less chance of pneaumonia or spasm reactions to overly cold winters-morning air.

It's more difficult to breath through the nose, so you get 2 benefits here, firstly you wont hyper-ventilate, and secondly you will develop the strength of your diaphragm, allowing easier breathing next time out, and possibly a better singing voice.

Don't try breathing in through the nose and out through the mouth while swimming.

Don't know about fitness - I'm not very, but I train actors, who have to deal on a daily basis with large, raw emotions.  When they have been through a particularly emotional piece I always tell them to breath slowly and deeply in through the mouth, then expel the breath in a deep sigh out through the mouth.  I call this 'controlling breathing'.  It helps them relax and come safely out of the emotional state they've been in.   If they need to access such an emotional state I advise that they breathe in and out fast and deep through an open mouth.  I used to worry about hyper-ventilation until I found that its dangers are vastly over-rated.  A course in Primal Integrative Breathing - apart from dramatically lowering my stress levels - showed me the power of your breath.  This simple technique - consisting mainly of controlled breathing in and out through a wide open mouth (Circular Breathing) - used to be called Rebirthing back in the hippy days of the 60's.  Unfortunately some charlatans latched on to the more sensationalist aspects, and then some idiots managed to kill a poor young girl practising what they said was Rebirthing, but which, sadly, had little to do with a really valuable technique.  This technique is especially valuable for actors.  Their ability to access the full range of their breathing to support the voice is of paramount importance.  So, really, all the fitness instructors are trying to prevent is you getting over-emotional.  I've heard of the same problem/solution from Yoga instructors.

As a Physiology student and student of a martial art, I too was intrigued by similar advice from my martial arts instructor. My initial answer is that it decreases the amount of physiological dead space. Physiological deadspace is the volume of air which is inhaled, but doesnt reach the gas exchange tissue of the longs. Basically it is the volume of the mouth, trachea, bronchi and broncioli (sp). By breathing in through the nose, the 'dead space of the nose' is counted for breathing in, but when you breath out through the mouth, the nose space (and sinuses etc) sint counted, which means you can exhale a bit more air than if you breathed out through the nose. This gain is probably quite small though.

Secondly, since most people breathe automatically in and out through their nose, by changing it to "in nose, out mouth" the person has to think about their breathing. This helps control your breathing and slow it down, which is good for sports as it means you breath much more effficently and deeper, which gets more oxygen into your blood etc.

There is also the litttle (but rather important) matter of if you breath out through the nose, mucus might come out - particularly if you are doing intense exercise and sweating and panting, which means you probably are exhaling quite forcefully.