The short answer
Temperature falls with height. On an average day the air is about 6 °C colder for every 1,000 m you climb, and in a storm, when the air is well mixed, that rate is fairly steady. A resort with 1,000 m between its village and its top lift therefore spans about 6 °C, which is more than enough to cover the gap between rain and snow.
What that looks like in a forecast
Say a storm arrives with a freezing level at 1,900 m. In a village at 1,400 m the air is about 3 °C and it rains. At a mid-station at 1,900 m it is 0 °C and heavy, wet snow falls. At the top lift at 2,500 m it is about −4 °C and the snow is dry and fast. Everyone in the village complains; everyone at the top has a great day.
Because of this, the snowfall total for a resort is never one number. On snowingplaces each resort page carries a forecast for the top lift and one for the village, and the lists and maps use the top-lift figure, because that is where the snow is.
Which resorts suffer most
Low villages with high lifts see the biggest difference: much of Austria, the Portes du Soleil, the Dolomites’ valley towns, and the coastal resorts of the Pacific Northwest and New Zealand. High villages such as Val Thorens, Obergurgl or Breckenridge rarely see rain in midwinter because they sit above the freezing level of all but the mildest storms.
What to do about it
Check the freezing level in the forecast, not just the temperature in the village. If the freezing level is between the village and the summit, go up early and stay up. If it is forecast to fall during the day, the afternoon is better than the morning. And if it is above the top lift, it is a day for the spa.