Always consult the official Owners Manuals first
Comparing the Axe-Fx III to the Axe-Fx II
The information on this page supplements the official manuals.
Comparing the blocks
Comparison of available effects in each device.
Comparing sound quality
”Everything sounds better. I think a big part of it is the quality of the I/O. The I/O measures flatter and less distortion than my $2000 interface.” source
"It actually does sound a little better. The extra DSP horsepower means that we didn't have to make compromises in some of the algorithms. The amp modeling algorithm is very similar but there's a few places on the II where we had to make compromises to get the algorithm to run within the allotted time. Also the III has a higher internal oversampling rate and a higher bit depth on some calculations (64-bit vs. 40-bit)." source
"Better algorithms, higher upsampling, better analog I/O design."