I like the Razr screen but for everyone who likes it, there are those who don't. If you are going to use the phone outside alot, it is hard to view unless you have your brightness up - the Bionic screen is the better outdoor screen. If you have a local VZW store that has the phones (working) on display, I would go and check them all out.
Also, I don't think there is much change in the screen but the Razr Maxx is also upcoming for release (extended battery version)...
I use mine as a digital gauge cluster for my car sometimes (check out the app Torque) and it's often in direct sunlight; it's as good as you can expect a backlit, non-eInk screen to get. Very readable.
As to competitors, you have two if you're going Verizon: the Galaxy Nexus and the Rezound. I won't get into which is the better handset, but I'll simply compare the screens. Straight up, the Galaxy Nexus has the better screen. It's the exact same tech (sAMOLED pentile matrix) but larger and higher resolution and ppi. That said, I really don't think the difference is too noticeable apart from getting 2" from the phone and staring at static text.
The Rezound has a higher resolution, denser ppi display, but it's an LCD. It will appear crisper from 2" away, like the Nexus, but it won't have the contrast, deep blacks, or vibrant colors that the RAZR and Nexus do. I hear it's also less readable in daylight, but I can't speak for that.
Best thing, though, would be to head down to a Verizon store and see for yourself. Bottom line, though: you ought not be disappointed by the display. It's really pretty awesome.
When I looked at the Nexus side by side with the Razr, they both looked pretty good to me and nothing on the Nexus jumped out. I thought the Rezound looked good, but I wasn't really interested in it. What are you upgrading from?