Pat is right about hide glue sucking the rawhide down to the substrate. As it loses moisture it REALLY loses volume.
Still, I prefer to use TBII or III because it is tackier and less slippery. I find the rubbery wet rawhide is about as hard to herd as a flock of cats, and with slippery hide glue beneath, it becomes a chore I just do not wish to take on. The aliphatic wood glue (TiteBond and other most other wood glues) gets sticky faster and still has enough work time that I can massage out all the air bubbles AND excess glue. Trust me, those two issues are more important than glue choice.
Both will work great in the end....IF you have no air bubbles or unsightly clots of glue. Good luck, and as always, post pics of progress for folks like me that can't read.