There are some good comments from the piloting side, but there are other factors involved as well. From that ATC perspective though:
Airports and the surrounding airspace can only handle a limited amount of aircraft before things start to get dangerous. When weather conditions are less than optimal, the number of aircraft the system can handle decreases. For example, an airport may be able to safety handle 60 arrivals an hour in good weather with no risk to safety. During a snow storm, poor visibility may reduce that number to 40 aircraft (or less) per hour. The reduction in capacity means those aircraft may have to wait at their home airports, connections get missed, and generally fewer airplanes can go flying. The last thing anyone wants to to have more airplanes in the air than can safely be allowed to land. So the delays are shifted to departing aircraft, which often translates into flight cancellations.
TL:DR The amount of airplanes an airport can safely handle decreases on a sliding scale as weather conditions deteriorate. This means some flights get cancelled and delayed while others do not.
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