Hans, you need to think about the single chain from start to finish in a logical manner. You plugged your guitar into the Axe II into the Twin. So let's break that down. First, it's guitar to some number of virtual stomp boxes to a virtual preamp/poweramp and cab probably followed by some combination of reverb/delay/etc. and then into a real preamp/poweramp and speakers. So to extrapolate that into physical devices you plugged into some pedalboard then into some preamp/poweramp that had a line out (or you grabbed the line out from a load box) then into something like a Palmer DI or H&K Redbox then into a digital delay/reverb and finally into your Twin. That'd be a less than ideal physical signal chain so it's not going to be a very good virtual signal chain.
The Axe II can be used in many ways and configured to do lots of things. You unfortunately chose the least optimal usage of all. Assuming you want to use the amp modeling in the Axe II, here are some very good ways to hook it up:
Axe II -> FRFR speaker - this is essentially running direct and you would run with power amp and cabinet modeling enabled on the Axe II. This should be a PA power amp into a PA speaker or a powered PA speaker. The sound coming out of a system like this is like listening to a mic'd guitar rig. It does not sound/feel exactly the same as standing next to your tube amp and cab - it's more like playing on a big stage where you're relying on what's coming out of the monitors more than your actual amp rig. This takes some getting used to whether it's modelers or real amps that you're using.
Axe II -> PA power amp -> guitar cab - in this mode you have power amp modeling enabled but cabinet modeling disabled on the Axe II. This mode works great if you really want the amp in the room feel but you want to have the full amp modeling experience.
Axe II -> guitar power amp -> guitar cab - in this mode you have power amp and cabinet modeling disabled on the Axe II. In terms of amp modeling, this is basically using the Axe II as a preamp. For the guitar power amp, you can use something like a Boogie 2:90 power amp or the effects return of a good tube amp.
I've gigged in all of these configurations. The best on stage feel/vibe was running the Axe into a guitar power amp and cab. The best the band sounded was Axe running direct because we could control our on stage sound much better which means you have more control over the FOH mix.
I gigged with the Ultra then the II for 3-4 years and loved it. I basically had 4 patches - clean, crunch, bunch-a-crunch, and balls out crunch. I spent the majority of the time using the middle two patches. I play with my fingers as much as with a pick and I use the volume control and pickup combinations a lot. The Axe responds to those nuances brilliantly - it never neutered a guitar ever.
Bottom line: until you've experienced the Axe II (or Kemper or any other modeler) in the right setting you really aren't getting a good idea of what it can do.