In 2016 I bought a Kemper and coming from earlier effects units, it was a revelation soundwise. I knew about some of the drawbacks (such as no editor, clunky interface...) but thought that tone should be king. Well, I didn't use the unit as much as I thought I would. The lack of an editor and the fact that there's no easy USB connection to your computer were more of a dealbreaker than I had originally anticipated. Too many cables necessary for recording (plus one extra interface).
The feel of the unit was great, no question. I was at first a bit disappointed by many of the presets, though. It's like others have said before: You really need to go and find rigs that fit you and your playing style - that's the part I didn't enjoy. Of course, I went ahead and bought commercial rig packs, but the amps therein all sounded same-ish (because they were most likely captured through the same chain, just with the amps exchanged).
I find the Kemper more suited to live playing compared to home or studio work. Some guitarists don't care much about which gear they play through (and still sound great!), which I envy. For them, it could be as easy as dialling up that M.Britt Marshall, add reverb and be done with it. Being more concerned with tailoring my tone, the Kemper didn't deliver in that aspect. With the Axe FX III, I feel like standing in front of a big shelf with all the amps, speakers and effects in the world from which to choose a tasty combination. With the Kemper, it comes down to endless rig and effect browsing with a clunky interface.
So, usability and what you actually need the unit for is a big factor, in my opinion.
But since you were asking about the tone: I really liked the clean and driven tones. Many times you can read this is something the Kemper excels at. I won't deny the fact that the Fenders and Voxes etc. sounded great. However, with gainier amps, I felt there wasn't so much difference between actual models and there's always this compression in the mids, which was mentioned here, that at first makes it feel super-fluid under your fingers but is ultimately not quite realistic. Also, over time, I noticed a cocked wah tone in most of the higher gain models that I couldn't dial out or unhear. On the Axe FX III I was floored by the fact how different the amp models feel and react under your fingers. To my ears, the tone is more open with better high-end extension.
When I got the Kemper, it already felt like a unit in the latter part of its lifecycle. Years earlier, it would have wowed me to a much greater extent, I'm sure. The Axe FX III to me is just the next level in evolution. I didn't appreciate the general opinion on the Kemper forums that the product has been created more or less perfect, proven by the fact it's basically an unaltered unit since 2011 with no need to go any further than that, being rather complacent and smug about it. I believe that technology needs to progress and the boundary needs to be pushed, like the Axe FX III does. How is it possible that in 2019 despite some substantial Kemper updates (delays and reverbs and such) there's still no editor? It's coming soon (tm).
I got the Axe FX III in autumn 2018 with the firmware 1.18. Now we're on 6.04 with substantial and clearly audible improvements to the amp, cab, drive and effect blocks and much more. That to me is the biggest difference.