I've been over Hwy 14 and 16 dozens of times each -- maybe hundreds of times over 16. I recommend Hwy 16 out of Buffalo to those wanting an easier route. For scenic beauty, 14 and 16 are close. I think 14 (out of Ranchester/Dayton) is a little prettier going up and coming down, but 16 is prettier across the top. 14A (alternate) is very scenic but a little steep with sharper curves, and I'd not recommend it for towing a long 5er. I'm sure you could make it, but I'd not recommend it. Time-wise, there's not much difference in any of them. I once had a business partner from Greybull who alternately drove 14 and 16 to Gillette. He said one was 5 minute quicker, and I think that was Hwy 16.
The past couple summers Hwy 14 has had some detours and closings due to heavy rains washing out the road and causing rock slides. Nothing yet this year that I've heard.
More often than not, when I go to Cody (from Gillette) I take 16 to Worland, then head through Greybull to Cody. I take 14 now and then for a change of scenery.
If you'll be stopping at Devils Tower (the country's first national monument), there's a couple good campgrounds there. We prefer the one inside the park -- Belle Fourche Campground. It's got no hookups. If you want hookups, the KOA just outside the gate is nice too.
Buffalo has a few nice commercial campgrounds in town -- KOA, Indian Campground and Deer Park Campground. Or if you want something in the mountains, Tie Hack Campground is 12-14 miles out of Buffalo just off Hwy 16. It's one of the newer NFS campgrounds on the mountain and was designed for the bigger rigs. No hookups, just spacious campsites carved out of the pines with picnic tables, fire pits and the occasional deer or moose.