The AX8 is great...I'm still using mine live and I can tell a difference with the new firmware updates...esp on the fenders. They seem more responsive, def more like the III. BUT the III is completely different to my ears. The twin sound is like that good old fender twin that you never want to get rid of. Both do the job.
For 1100 bucks, the AX8 is more than worth it.
The biggest issue I run into with the AX8 is cpu. I like to have 1-3 presets I use ( a fender based one, a dirty one, freidman, marshall etc, and acoustic one). I very often run into cpu issues, mostly with the fender one.
The one main thing to consider with both, to me, is the CPU, not so much the sound. I am going to hedge that if you heard me play live or on a recording you couldn't tell which one I was using. With all the sound that goes into it, unless you are side by side comparing, you may not miss the III - if you never had it. The ax8 is really that good,more than enough.
With the III you get it all. Two amps at once, all kinds of stomp options, the newest technology...a big seller, and enough CPU and the best sound.
I will add with the II I had almost as much CPU issues, not as many, but enough to where I sold the II and kept the AX8 with the III, purely for flexibility.
If I was a "stay at home" guitarist comparing 1100 to 3000 (the III and a FC) it kind of is a no brainer what to buy.
If I gigged or recorded, even if once in a while - no question the III.