The Appalachians are snow covered because they are still higher than anything around them. While the old adage is that “hot air rises”, that’s only true to a point. As the previous answer explains, upslope flow is responsible for this in addition to the cold air that blows in from the Great Lakes and other water in the area.
From there, it’s the basic water cycle. If the water is above the Appalachians when it’s ready to fall, it’ll fall as rain in the summer and snow in the winter. That’s about the only difference there is to how it falls up there. That’s also why the Appalachians still get snow, despite the fact that they are not among the highest mountains around.
There is an upward flow of air towards the higher terrain which has the effect of forcing it higher still and over the wind-swept slopes. This is called upslope flow. Any moisture in the air will form clouds and may cause thunder in summer and snow in winter.
As winds pass over the water of the Great Lakes large amounts of moisture is carried upwards towards the mountains. The climate is temperate with a clear difference between winter and summer. There's constant risk of frost, even so.