Mid control has the biggest difference in overall tone I've found, really seems to shave the fuzz the most. I've also had pretty good luck putting a mild OD after the fuzz (you can run 2 drive blocks in a row, FYI) and this shapes the tone a good bit too, EQ block would also work.
I've got a Wampler Velvet Fuzz sitting right here and can pretty much copy the tone when I a/b it, but only for a given knob setting. Adjust anything else, even the pickup used, and it goes out of whack and I need to adjust again.
Its really just a complicated mix of EQ, high and low cuts, drive, and output levels more than anything. Sometimes the clip type makes a difference. Using the Octave fuzz and changing the clip type from recto takes away the octave up effect and sounds pretty good as a general fuzz (Band of Gypses fuzz is octave up minus the octave part of the circuit from what I understand).
The overall level of the amp makes a difference too
The Wampler Velvet kind of mimics a cranked amp a little bit too, so to cop it closer you need to run the drive block into a slightly dirty amp....
See thing to remember is that in the real world, most people can't always run a fuzz into a cranked Plexi due to volume issues. With the Axe, you can easily emulate the tone of an amp cranked up loud, and the way the fuzz reacts is really going to differ based on the amp model.
I find ones with a fair bit of headroom can work really well, HiWatt for example
Finally, the Velvet is kind of a dark pedal, even with the tone knob all the way up. Really have to cut some treble/high cut on the Axe to match it, as otherwise the drive block can sound too bright
As you said though, the drive blocks are pretty tricky..... The Axe can do a fantastic fuzz tone, but it also does a bunch of fuzz tones that sound like crap unless they are carefully tweaked, which can be a bit annoying to say the least.
The tones are there though......
I've a/b'd and sold off my Axis, London I and II, Fulltone 69, Dunlop turqouise, Si, Ge and BoG Hendrix FF's, Lunar Module, P19, Box of War, Big Green Russian, If 6 was 9 fuzz etc. The Axe really can come 99% of the tone to where I couldn't really tell them apart, but I will have to sit and go back and forth for like a half hour before I can match the tones