I Understand things slightly more operationally in a question of certification. Maybe i'm wrong, but you live and learn.
Cat IIIa 50'/200m so on the roll out the captain can keep the aircraft on the centreline using rudder control as soon as the a/c completes the flare, autopilot has to come out, as there is no rudder control by the autopilot. If you leave it in the autopilot tries to maintain the centreline using aileron, which on the ground is useless
cat IIIb the autopilot can control the rudder and so can maintain the centreline during roll out. Hence lower minima.
cat IIIc the autopilot can complete the roll out and taxi the aircraft to the stand. (which at the moment is impossible as there is no suitable guidance systems). if it's 0/0 you can't see so you can't manually taxi.
I'm willing to be corrected, but this is how i understand the essential differences between the 3 options. I've only ever flown cat IIIa (i really don't like it btw, very unnerving).