Main things for me is you can run dual amp rigs, and you can route more complete routings; typically never having to disable your reverb due to lack of CPU power, and last but not least, you can use it as an audio interface so you can record DI and effected tracks straight to your computer with just a USB cable, which I find really useful.
Downsides, you should realize, are that you'll ideally also want a foot controller, which adds costs, and then you've lost that great AX8 form factor.
Taking money out of the equation, I'd probably NOT buy a II if I had an AX8 already, and would move to to the III, if in budget, since that is a major step up, over both the II and the AX8, and will continue to evolve for years being the latest flagship product.
Might still want an AX8 for smaller gigs even with a III though..... I don't gig really these days, So I sold mine. Loving the III