To air layer a tree I cut all the bark of in a strip around the branch and surround the stripped bark location with moss or some other soil substitute. It seems I could just cut the branch off at that point, stick it in some rooting medium and produce exactly the same result. If I have a cutting it seems it would be much, much easier to keep the rooting medium damp than trying to keep the air layer ball damp.

What factors would make me choose one over the other?

There is a limit to how big cuttings is possible to root. For air layering there are no such limit. Use air layering for thick branches.