"The snow is so heavy..." sounds as though the snow itself is too heavy to lift, and no that a lot of snow has fallen. "It snows so heavily..." sounds like a general thing, like where we live it snows often. I think I would say "It snowed so heavily that we had to stay home. Or we are staying home because it snowed so much. I think you have to make the snow and staying home more of a cause and effect thing.
Hope that helps!