It has to be cold for it to snow, if your definition of cold is like that of most folks who live in the mid-latitudes. Therefore, significant snowfall at such very low temperatures is rare.
Most clouds form through a process called expansion cooling.
Expansion causes the air to grow cooler, which reduces the amount of water vapor it can contain. The 'excess' vapor condenses out into a cloud. At very cold temperatures, the expansion cooling process starts in air that already has a low water vapor content. At higher, but still subfreezing, temperatures, ice crystals hook onto each other to create snowflakes. In extreme cold, the ice crystals remain independent.