This was a real tough one, because I really love The Exorcist III for all the reasons you mentioned and two years ago I would have chosen this one, but after re-watching The Exorcist a couple of times over the last couple of years it now is one of my all time favourite (horror) films. I'd like to point out one aspect of The Exorcist that you didn't discuss during the episode and that is the use of music which is fantastic. The snippets of modern classical music - especially Penderecki - make it even more creepy and I wonder how much Kubrick was inspired by this for The Shining. The most famous piece of music is of course Mike Oldfield's Tubular Bells and it is interesting that for lots of people - including myself - it is viewed as one of the most iconic pieces of music in horror cinema, on par with Halloween and Psycho. Yet ironically, Tubular Bells is not original film music and it is only used twice in the entire film and for a very brief period of time at that (excluding the end credits).
So both are fantastic horror films in my opinion, but The Exorcist is a real masterpiece and one of the most important films of all time, so my vote goes to Friedkin's film.