Just ran across this thread. I see that most are southern so the weather part of my post won't matter, but I wrote it in case anyone comes from the east / Midwest.
We're from Kansas City and have been going to Moab every year since '05, twice, once in the summer and again the the week between Christmas and New Years. In all those years it was only snow free once, in '05. In '06, we did Top of the World in a month old stock '07 JKUR and there was around 5" of snow at the top. It's a lot rougher nowadays. They had record snow a couple of those years and we spent a lot of days doing very little. Every other year it's been cold enough to where we couldn't (safely) do some of the more popular trails due to parts of the slickrock being snow covered. We slid down a steep part of Fins and Things 2 different years before I finally said 'no more' slickrock if there's a chance of snow on it. Both years most everything up to the icy spot was snow free and then . . . you're kind of stuck. Due to all the snow we changed it up from wheeling the harder trails to hitting all the easier ones and doing more site-seeing and even then some of those easy trails go across open land so there may be 12"-20" snow drifts to deal with. That happened on, I think, Picture frame arch. We've always had JKUR's. A '07, a '12 and now a '14. All with bumper / winch, 2.5" lift, 35s. etc. As was said, it's way different in the winter time as 95-99% of the time you won't see another rig on the trail so plan ahead and make sure you're also able to deal with ice. (We started bringing ice melt.)
For anyone from the east / north / midwest. . .
Keep an eye on the weather every day. On the day we were supposed to leave (back in '06) a storm was coming but since it was supposed to hit late in the day we went ahead and did Chicken Corners (10x more fun in a side-by-side). By the time we were done the storm hit and we spent all day stuck on the highway only getting to Limon, CO where they closed the highway and we were stranded for 3 days. Another time we didn't pay attention and our 13 hour drive home ended up taking 27, which I drove non-stop to beat the worst part of a storm. (Thank you 5 hour energy's!) During that trip I drove from Moab to past Salina, KS before it stopped snowing.
I'm not trying to detour anyone as Moab is great anytime and, imo, is way prettier in the winter when there's some snow around but just want to make sure anyone that hasn't been there in the wintertime is prepared.
Oh, as for where to stay, make sure to check out VRBO as they've built hundreds of condos in the last couple years. During the winter it's a cheap way to stay in a nice place with a fireplace, 2 car garage, etc. We usually stay on the south side of town up close to the KOA campground. It's close to Susie's Branding Iron, which we usually go to for dinner. (Usually the food is great, especially the Chicken Fried Steak, but it sucked so bad last year we didn't even go there this summer.)
Anyway, sorry for the long winded post, just trying to help!