I fly very frequently and have flown in and out of snowstorms several times.
Yes, flights to get cancelled, but unless the storm is significantly severe, it’s purely because of capacity at the airport. In freezing temperatures, planes must be de-iced before they take off, and at busy airports with say, 20 departures an hour, they won’t have 20 machines so that each plane can still get away on time. They’ll have 2 or 3. De-icing takes about 10 minutes (plus you need to queue for it), and if you only have 2 machines available for a busy hub - looking at you Frankfurt - you are obviously going to reduce your take off rates simply because there’s no capacity to de-ice everyone on time.
When this happens, they start cancelling flights - mostly short-haul - purely because the delays would make it impossible for the plane to come and go on a reasonable timeline so they “consolidate” many flights to at least have an attempt at getting some out on time.
Delays and cancellations can be expected. However, if the airport is prepared, has cleared runways, the air warms and you’re flying out of a smaller airport not a hub, your chances are good at making it out. It’s just an unknown…