what is the difference between "apparent" azimuth and elevation? and non apparent azimuth and elevation? Is there even such a difference and how do I convert between them?

Since your location on a planet can either be in planetcentric lat long coordinates or planetographic lat long coordinates i would think that would effect your azimuth and elevation locations coordinates for the sun but not sure. Its confusing to me.

the internet is very inconsistent when using the word 'apparent" when defining azimuth and elevation.
Not mentioning the word apparent might mean they are assuming "apparent" but Iam confused.
thank you.

If you're discussing remote observations of astronomical objects, "apparent" includes various aberrations that affect how it would be seen vs. the purely geometric solution.

More detail here at JPL Horizons documentation for how they calculate apparent positions.
http://ssd.jpl.nasa.gov/?horizons_doc#specific_quantities

The key bit for Az & El is as follows:

Just one evocative example of this that I've read about recently… the southerly declination of Canopus plus the northerly latitude of Lick Observatory add to almost exactly 90°, but observers there have reported seeing Canopus at about 1° altitude. That is a case where the difference between apparent and ordinary azimuth makes the difference between seeing something and not seeing it.

Indeed refraction is nominally about 1/2 degree for something on the horizon if you're at sea level. It can be more if you are on top of a mountain looking at a horizon at sea level. On top of that the horizon itself is depressed downward when on a mountain top.

NOTE: Edited the topic title for brevity.