Difference between OBEs and APs?

What is the difference between OBEs and APs?

Just a slight difference in terminology. OBE is a more general term, essentially any experience in which one's consciousness is operating outside of the physical body's limitations. AP is more specific, an OBE to the astral dimension.

There are different kind of dimensions you can project into, they differ by the state of consciousness you need to be to exist in... and how the surroundings you are in looks differently too. Astral dimension is lowest one...
As Vince said, OBE is a general therm of operating out of body (outside of physical body).
Main connector is that you know that your body is in bed, you know that you aren't dreaming, that what you experience is real. I read explanation, that this is because our brain is confused and center active when one is awake is working literally overtime, much more actively than in normal waking time.
When you are experiencing lucid dream you know that your body is in a bed, but you know that you are dreaming the center in your brain which is working while awake is on low activity in comparison to normal waking state.

Whether OBE is only special case of dream or real thing... well science tends to describe it as special case of dream. Subjectively it is spiritual experience. I try to find connections to reality and... well I don't have proofs good enough to tell whether it is really spirit moving out of body or remote viewing from out of body perspective, or just fantasy. But that shouldn't be discouragement to any explorer, isn't it so?