PTO power take off, used to drive rotary machinery using engine power. You only have one if you have a hydraulic pump, blower or other piece of equipment that needs to be powered. Usually you don't leave it on but only engage it when you are loading, unloading or doing something that needs these special functions like raising or lowering your dump body or starting a blower to load, unload a hopper body. In other words you don't turn it on and leave it on.
The axle diff is to be engaged while the wheels aren't spinning out and not to be used above 45 MPH. It's a very small gear the engages the other diff.
The trainer told you wrong It is only use for traction when you are stuck in mud or snow or ice in some situations when you have a heavy enough load. It might not help you if you are empty. Not for highway use. If you need more traction when you are light, you will have to throw the chains on if you have them available. Unless you are sitting deep in a mudhole then you will have to call a tow truck.
