For a no compromises unit, you can’t beat the Axe3 with an FC controller.
I own both. Here are a few differences between the two off the top of my head:
The sound quality, feel, touch responsiveness to both units is excellent. The FM3 is currently a FW version of two behind the Axe3 in development, but presumably will catch up soon.
With the FM3, you will run into the CPU limit if you want to built complex or kitchen-sink presets. Reverb in particular is a CPU hog in high-quality mode. If the fine details of reverb are important to you, the FX3 added an ultra-high quality mode a while back that IIRC will not be ported to the FM3. Only one reverb block in the FM3 so transitioning from a long lush reverb to a shorter reverb time is not a smooth transition like it would be in the FX3 using two reverb blocks. If you like to stack drives, using two drive blocks on the FM3 takes a fair amount of CPU as well. With the FM3, I have to pick and choose which blocks I want in a preset and I don't add a lot of extra effects that I "might" need. I tend to build per-song presets anyway so this is not a problem for me. Both units have 4-channels per block which means that you can set up four different settings within that effect type. You can then switch channels using scenes or assigned switches. So switching, for example, from one drive type to another, or from a clean amp channel to a gain amp channel, does not take two blocks in the preset.
If you want to make your own IRs, the Axe3 is necessary. The audio interface capabilities is different between the two units. And of course the currently shipping version of the FM3 does not include the headphone jack. If you are not on the FM3 waitlist already and are just signing up, by the time your name comes up this will probably not be a problem. That's another factor, the Axe3 is available now but the FM3 is currently a waitlist purchase that could be a few months or more before the list is cleared. No one know the answer to that.
If you have the budget, my suggestion is to go for the Axe3. In my case, I added the FM3 because I play weekly at Church and I don't like carting around a rack with the Axe3. For what I do in a live setting, which doe include a fair amount of ambient sounds, the FM3 has enough CPU, sounds excellent, is portable, and will serve me well.