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. There actually is no such thing as too low a temperature for some sort of ice crystal to form and for such crystals to settle out and land on the surface.