Three factors affect Cat III minima for any individual flight:
1. The technical capability of the equipment installed in the aircraft
2. The operating minima set by the aircraft operator, this may vary by aircraft type and can be more stringent than 1 above (i.e. they deliberately give themselves a safety net above the technical minima)
3. The training certification level of the operating crew - if they ain't both Cat III trained they ain't Cat III landing