I own both and love both. They each have strengths and weaknesses.
Personally, I use the Kemper live in a cover band setting and the Axe-Fx for home recording and video work. I like the portability of the Kemper and the "real amp" adjustability in a live setting. It's easier to tweak on the fly compared to the Axe-Fx, plus I have the Powerhead, so I don't need to have a rack with a power amp plus the Axe Fx. I also like how the Kemper allows you to sum outputs to mono, since most PAs I play through are configured for mono. Also easier to use a real cab on stage and send a cab sim signal to FOH.
This isn't to say that the Kemper isn't a great studio tool and that the Axe Fx isn't viable in a live setting. But for my needs, Kemper/Live and Axe-Fx/Studio is the way to go.