I can only echo what the others are saying. We don't have an Axe but I would be shocked if you could not get any tone you want out of the Kemper.
From what I gather on this forum the difference is the workflow. Don't underestimate the workflow! If you get enjoyment out of tweaking options to make things sound even better to you, probably the Axe is what you want. If you want a few dedicated knobs and buttons and don't want to descend into menus to change every aspect of the device, the Kemper is hard to beat.
For us - I play with the kids at home - it is turn it on, while waiting for it to boot tune up and then let's play. We rarely touch the device, just change from rig to rig and maybe turn the reverb ad delay knobs a little.
The laptop that I have connected to the Kemper has Waves Tracks Live running, recording both the DI and the "real" sound. It is extremely convenient to record every noodling we do and can re-amp in seconds if we don't like the sounds.
Monkey_Man, that picture with that huge screen looks photoshopped - like those advertisements of backyard pools with mini-people swimming inside.