I have an Axe III and Kemper. Both sound amazing with my headphones (Focusrite, came in an interface bundle). I used to hate using headphones, but in the past few months I've had to adapt and TBH I often prefer the sound with headphones over my Adam A7X monitors. I'm quite introverted and don't particularly like others hearing me play, so I think there's likely a psychological aspect there too.
I love both the KPA and Axe. KPA sounds slightly better through headphones IMO.
Amp tones can be dialled in to sound stunning on both, but the feel is a bit better on the Axe.. The Axe has better effects, but I fell in love with the Kemper 'Ionosphere' reverb. To me, it's the best shimmer reverb I've ever heard. The reverbs and delays on the Kemper are outstanding IMO, their pitch-shifted delays and reverbs are the best on any device. Not sure what magic Christoph worked, but getting reverbs and delays that good on a 10 y/o device is seriously impressive. The Axe has better 'conventional' reverbs, and having the ability to use parallel paths is a godsend. The new drives and fuzz are good in the KPA, but I wish there were specific models like in the Axe. The drives on the Axe are much better, IMO. As good as the real pedals, definitely. Better really, as they can really be tweaked. I think Cliff said he would love to bring out a FAS drive pedal to put all these boutique builders out of business. I think he definitely could.
Cygnus sounds spectacular, and I think the Axe III currently has the best amp sounds out of everything currently available. Absolutely amazing amp tones. I also think the Axe III takes pedals really well, better than the KPA. I definitely use the Axe III more than the KPA, but they're both amazing tools.