snow is generic.
My children adore snow (in any and all of its manifestations).
the snow is definite.
My children adore the snow (that fell
last night), the snow (that we have this winter), the snow (that they
play in), the (activities that they enjoy in the) snow, i.e.,
the snow activities, the snow play, the snow experience.
That said, the qualities that define snow in
the snow (
as used in one out-of-context sentence) are so vague that even
the snow seems generic here.
CJ