'Snow' is what is called a 'mass noun'. Like 'water' or 'traffic', it already describes a collection of things (a mass) and so there isn't likely a number of such items, just some of it.
'The snow that fell overnight was deep' is correct.
'The snows that fell overnight were deep' is incorrect (or poetic)
You can however talk about multiple instances of water, 'the glasses of water' which can be said grammatically as 'the waters in those glasses' but that is a strange transformation, almost poetic, and surely to be avoided in speech and most writing.
The same goes for 'snow'.
'The snows that fell in each of the towns were deep.'
works and occurs, but is a somewhat strange way to say it. One would more likely reword it to avoid the strange plural.
'The snow that fell in each of the towns was deep.'
or to emphasize that a snowfall occurred in a number of towns, and though the depth was different in each one, the amounts were all deep.
'The different amounts of snow in the towns were all deep.'
sounds much better. So the answer is yes, 'snows' can be used as a plural noun but it is not common and is not natural sounding. So if you are translating from a language with plural snow, you should reword to avoid the plural.
For a comparison here is the google nGram comparing 'the snow' and 'the snows':