There are several reasons for the use of snow as a motif. The first is its association for purity (`pure as driven snow`), an association that dates back at least to Shakespeare who associates snow with purity in Macbeth and A Winter`s Tale.
As well as being pure, snow represents a force of unregulated nature that does not exist the regulated world into which Jonas (cf. the Biblical Jonah) is born. Snow symbolizes something that can be painful (frostbite) and inconvenient, but still beautiful and exhilarating (as in the sledding memories). Thus it symbolizes that the community`s attempt to remove pain also removes the possibility of joy.
The children in the society are promoted to the next age bracket in December. In a free unregulated society, there would be snow at this time and so the absence of natural snow signifies the absence of normal aging, replaced by conventional age brackets, just as natural weather has been replaced by artificial uniformity.