V/S
The way I understand it is:
737 (classics) have two autopilots, able to monitor each other.
A320's have two autopilots, each with their own monitor channel.
737 : On a Cat IIIa approach the 737 Cl has an autopilot flying and the other monitoring. In the event of a failure, the system disconnects as there is then no integrity monitoring occurring for an autoland to take place.
A320 : In the event of a failure being detected hands over to the other autopilot. As this A/P has its own monitoring channel it can continue the approach to autoland. Hence this system is Cat IIIb. Each autopilot is capable of a Cat IIIa approach independently.