Physical exhaustion to me is my frequent 5mile run, or 2hours of straight biking ~15mph. It has to do with the body feeling sore, unable to move for the time being.
In mentality my brain is still going. It's just my body not being able to keep up with my brain. However physical exhaustion has one big plus. It trains your anaerobic productivity so that not only your muscles stronger, you can last without as much O2. Provided that you don't overdo it, you
always get stronger.
Mental weariness (omitting mood-related) to me is hours of reading up any moderate+high range of physics, Latin taxonomy, chemical properties and nomenclature, working at the company for a stint of 10 hours to get a project done; meet the quota or deadline of a client. Where your cerebellum is in charge and there are no pit stops waiting for you, leads to exhaustion.
But your body can get unhealthy from too much mental exhaustion. You only benefit in one area, in your head.
Moderation is key.