Here are the three layouts I used. I picked a mosfet symbol that showed the body diode as on the datasheet for the 2N7000 (the n-channel enhancement mode MOSFET I used):

As you can see Mosfet A uses the diode to block the body diode. This is the set-up I expected to sound the most different. I expected Mosfet B to act simply as a diode with a higher forward voltage. I used the leds as a control for comparison (the forward voltage was fairly close). My results surprised me!
First, here is a comparison of each of them used to rails after a simple Pro-co RAT clone op-amp stage:
NOTE: the board style seems to be cutting off the far side of the spectrum analysis (where the most intersting stuff is). If you want to see it just right-click then "open in new tab"
LED:

Mosfet A

Mosfet B

As you can see, the Mosfet A (blocking the body diode) and the LED to rail are almost the same. Almost! The spectrum analysis shows a little difference, and the difference was barely audible. I expected Mosfet B to act like that though, whereas it created a much more different sound. A noticeably different sound actually.
This stands out much more when I put then in the negative-feedback loop of my op-amp:
LED (op-amp feedback)

Mosfet A (op-amp feedback)

Mosfet B (op-amp feedback)

Here you can see again that the LED and Mosfet A are close. The difference is more pronounced this time however. Mosfet A is definitely a bit more aggressive and brighter. Not in a bad way though! Probably not enough to be actually worth it though. Mosfet B on the other hand produces something noticeably different. It's got a lot less high end, and it's generally smoother.
So, my question is: why? What is actually going on that creates this behaviour?