Getting back to the OP, I think a lot of us seem to be dancing around the same answer.
Assuming that during fog we’re using only ILS (Instrument Landing System) approaches*, there are three main styles: Category I, II, and III. Cat I approaches are good down to 1800’, Cat II are used for less than 1800 but more than 1200, and Cat III are good from less than 1200 to as low as 150’.
The problem is that the components on the ground to support the ILS (a localizer, glideslope, and DME) are expensive to operate and maintain. The more precise the system is, the more expensive it is to run. Dorjän’s first link gives a list of all of the Cat III ILS systems in the US. Notice that there aren’t a lot of them - an ILS costs money, and no one wants to spend it unless they have to. Boston-Logan only has one Cat III (Runway 4R), even though it’s a foggy airport. The same with Providence** (Rwy 5). There are other ILS approaches at those airports, but they are only Cat I systems. Even so, not all of the runways at an airport even have an ILS approach. Boston-Logan has no ILS at all for 4L/23R, nor Rwy 16 at Providence.
Boston-Logan makes a great example. Normally, during calm winds, there are three runways in use: 4L, 4R, and 9. Takeoffs are on 9 (and sometimes 4R for heavies), landings on 4L and 4R. If you can land an aircraft every 2 minutes on each runway, that gives you an acceptance rate of one per minute (2 mins / 2 runways = 1/minute). But remember, 4L has no ILS, so if it’s foggy, only 4R is getting used. That cuts you down to one arrival every two minutes, which is half of your normal rate. Half of the aircraft landing in the scheduled time creates backups. The rest of the air traffic control system will stop aircraft who are flying to the affected airport from taking off, which called a ground stop (or ground delay). Aircraft that were supposed to be on the ground at Boston are now still on the ground in Newark or Charlotte. If they’re not here yet, the passengers can’t get on, and it’s a delay. Aircraft already in the air can be slowed or put into a hold to wait for their turn in line to land. Using a portion of the normally-used runways creates a bottleneck in the system, which creates delays.
Snow does the same thing. The airports can’t plow everything at once, so the primary runways get priority. This reduces the acceptance rate, which backs everything else up.
The FAA has an online tool to show what the current problems are in the ATC system. If you go to National Airspace System and click on “NAS Status,” it will display the airports currently having issues - updated every 5 minutes. The “AAR” column is the Airport Acceptance Rate, which will drop for low visibility, storms, or snow/ice.
- There are many runways that have an RNAV (GPS) approach, which doesn’t use any equipment on the ground and is much cheaper. The FAA really likes RNAV approaches.
**Full disclosure: I work there