SLF,
The short answer is, It's not the quantity of the fog, but the quality !
Generally speaking for a Cat 1 ILS approach, there is a visibility requirement of around 550 metres ( this figure may be subject to some adjustment for ground lighting etc). Provided this visibility is reported then the Crew may commence an approach or perhaps more accurately may continue the approach beyond the final approach fix or equivalent position. Of course upon reaching the Decision Altitude usually around 200'agl they must have the required visual elements to complete a landing ( manual or automatic). If they do no, then they must go around and decide on the next course of action.
Cat 2 and Cat 3 enable a suitably equipped aircraft at a suitably equipped airport to operate in reduced visibilty with regard to the above requirements. Cat2 and Cat3 usually require ( not always) that the aircraft is carrying out an automatic approach and landing. Cat 2 reduces the visibilty requirement for passing the final approach fix to around 350 metres( measured) with a reduced Decision height of around 130 ft. Cat 3 (there are 3 sub categories) further reduce the visibility requirement to around 75 metres ( theoretically zero) with a decision height of around 14 ft.
It has to be said that whatever the reported visibility,it does not mean that upon reaching the decision height you will necessarily be able to achieve the required visual cues to complete the landing. In that case around you go (even though the aircraft may actually touch down in the process)!
Even on an aircraft equipped for Cat 3B conditions (14ft DHt and 75 metres measured vis), if the airport has degraded facilities and there is a long list of what is required, a Cat 3b approach may be ruled out. Likewise any degradation of aircraft equipment would also have the same effect. On top of that there are certification requirements for the crew and currency considerations.
I cannot comment on your particular flight, but as you can see the requirements are fairly complex. If for example the aircraft is certified for Cat2 then 300 metres would not be acceptable for one airport but 350 metres at another would be ok. Both is pretty thick fog to me!
I have tried to keep this simple. Hope it makes sense.