It means "completely pure" or "untainted", usually in the moralistic sense. The snow is "driven" per the Old English meaning of "drifted" or "driven by wind", not to be confused with snow that has been "driven on" by cars.
The phrase has a Shakespearean quality, though it doesn't appear verbatim in any of his plays. In The Winter's Tale though, a street vendor sings a song that begins "Lawn as white as driven snow / Cyprus black as e'er was crow". ("Lawn", in this case, being a type of linen, and "Cyprus" a crape-like fabric. Still with me? Okay.) In Macbeth, Malcolm remarks "black Macbeth will seem as pure as snow".
In a nice twist on the cliche, 50s radio host Tallulah Bankhead once described herself as "pure as the driven slush". (Bankhead, by the way, was also one of Hollywood's first "out" actresses.)
Another great Tallulah line? "My daddy warned me about men and booze but he didn't say a word about women and cocaine!"