My two cents, having just returned from yet another trip to Southern Utah from my home base here in Salt Lake City:
More often than not, what is most necessary for driving in So. UT is ground clearance, as someone has stated, and occasionally the low range gearing that is most often only available on 4WD vehicles these days (old 2WD pickups came with these so-called "granny gears").
Of course, it simply depends where you go--and if you head to So. UT, you will definitely want to drive beyone the 99.999% of the roads someone mentioned earlier. Otherwise, you'll be following RVs driving well below the speed limit for miles on end.
Cathedral Valley Rd. is definitely not for low-clearance vehicles, especially if the Fremont River is running high at the point where the road starts off Route 24. Whether a high clearance 2WD can make it depends, but I would ask ahead of time for a local expert's opinion. I've done this in a Jeep, and there were many times I needed the low gearing my 4wd low range offered. But--a great, remote drive.
Shafer Trail can be done with 2WD and high clearance (Outback or better), at least the time I drove it. We did use low range at some points, though mainly not to stress the brakes. May have been overkill.
The roads to/from Lake Powell--Hole in the Rock, Burr Trail (or whatever the Burr Trail turns into), etc.--I have never done, but I understand them to be long and in need of a stout 4WD.
I would suggest, however, that it is dangerous to presume you are all set to drive the Utah backcountry so long as you rent a 4WD. Have you ever driven a 4WD vehicle? Do you know when and how to use it in 4WD low? Etc. The 4WD in a Safari van, by the way, is AWD, I believe, and NOT what you need to drive the "really bad" stuff. At least, I'm pretty certain their transmissions don't go low, so to speak.
Too, if you have 4WD, it does not automatically mean it can run a road marked for 4WD drive only. I just drove that back road from Route 12 north of Boulder into Cap Reef (South Draw Rd, I believe), and the stock Jeep Wrangler I once had would have had a tough time of it in parts, simply because factory clearance was not enough (not even with 30 inch tires).
And of course, it always helps to have plenty of water, food, etc. In fact, it would be prudent to have whatever you need to make an overnight of it in case something happens.
Sorry for the ramble, I just got off the road and am beat. Hope this helps somewhat.