I've had this discussion a few times with Joey, who first turned me onto the FatBoys. And therefore my opinion is somewhat biased by what transpired after that initial discussion. AXY pickups are somewhat like very good studio monitors... think; high-end Genelecs or something. Fatboys are more akin to a really big stage rig. Still clean, but more of the good parts. Too much for some folks. They are just accepting sooo much more information. I love 'em... they aren't for everybody, but give me two anyday, neck and bridge, I'll do the rest.
But wait, there's more...
Personally, I could hear differences in AXY pickups from different eras. That '81 Distillate I had... it had the glossy finished plain ones, and they were a totally different voice than the ones that came in the '85 fretless, which had the classic shell with ALEMBIC branding pressed in. (not gold embossed yet). Recognizing that there were some other factors in play; placement, wood, setup, even stuff that I couldn't think of, I got a gleam in my eye and swapped them out one time. What I heard from each pickup mostly followed it to the other bass. It was a better education for what pickup placement and wood did that the pickups, themselves. All things being equal, they were darn close, but those older ones still sounder warmer, even when in the newer bass, with more modern placement closer to the bridge.
Later on, a tangentially related set of circumstances left me with a homeless set of FatBoys. (I had them in an '80 Series I, if you're curious... and they were uh-MAZING) After that bass got shipped home for new Series pickups, I put the surrogate FatBoys in the '85 fretless, and WOW, was it a different thing. I kept them in there until I sold that bass last Fall, ultimately reinstalling its originals. (those Fatboy pickups now live in RV's Alembicized Starfire)
My Custom fretless 5-string has a FatBoy at the bridge, and a standard AXY in the neck slot. The vintage routing shoved those things all the way to each extreme, so the neck pickup is plenty warm, but I can still hear the difference between them. That bridge pickup is audibly glowing red-pink. You can solo it out, push the lows, roll off the filter to about 30%, and... there is nothing on this Earth that sounds like it. Boost the Q at your own risk. Weirdly, it sounds more like my PJ activator-equipped Persuader 5-string than either of the AXY-equipped Distillates. Maybe from all that maple and purpleheart.
However they do it Alembic really lands it with people like us. It's a little bit nutty taking things quite this far for generating a dadgum magnetic field... but... if you get it, then you just do, and you can't un-get it. The pickups are just the beginning... then the signal gets routed through some other pretty cool stuff nobody else can do like they do. Yeah, they do sound a bit different, and depending on how dialed-in you are, you'll get a preference for one or the other, or a combination.
I kinda' learned to celebrate/enjoy the difference.