I have owned both lenses,
and used the Af-S FX lenses on all of my DX bodies...D 50, D 90, and D 7100. The AF-S 4-5 -5.6 is a marvelous lens, and I love the crop factor. I've never had a problem with focus speed, or anything else. I bought the AF-P 4.5-5.6 VR lens (that would be the FX lens) after about 10 years of using the older one...critters, air shows, race tracks, people and what have you. I have found the newer lens to be slightly, but noticeably sharper throughout the zoom range, and the 4 1/2 stop VR to be an advantage over the AF-S's ~2 1/2 stops. At 81, I need all the help I can get!
If someone is on a budget, there are good AF-S 4.5-5.6 FX VR lenses on the used market for reasonable prices. There are other versions of the lens...Non-VR, and for the latest ones, 4.5-6.3, also without VR,(?) which are lighter, but said to be lesser lenses. Personally, I have avoided them. YMMV. Recently, with the Covid inconvenience, I got the urge to do something photographic! I got my lenses out of the bag and said ; "All right, you guys, we're gonna do some sharpness tests!"
"O. K.! Knock off the NOT AGAIN!" NO, 50mm f1.8, you cannot be excused! No star charts, Air Force or other paper tests! We have a set of brightly-colored curtains in the "office", with large flowers. Soooo..with my 70-300 AF-P...wide open throughout the zoom range...I backed off to what I figured would be a reasonable using distance. Looking at the 100% blow-up on my camera screen, I could count the threads in the fabric! Scientific? NO, but good enough for any of my purposes!