This is an excellent question! Of course, the snow seems to operate symbolically in this thought-provoking short story, but in my mind, it is inextricably intertwined with the epiphany that Gabriel experiences at the end of the tale. Having been told by his wife, Gretta, about her first love with a man called Michael Furey, he realises that he has never loved in the way that Michael Furey loved and that Gretta returned this love. Thinking about his aunts and their inevitable deaths, he realises that since everyone is going to die anyway, it is better to live life to the full, experiencing emotions richly and completely:
One by one they were all becoming shades. Better pass boldly into that other world, in the fully glory of some passion, than fade and wither dismally with age.
The snow, that begins falling in the final paragraph of this tale, thus represents the universal nature of the human condition--that we are all "dead" in one way or another. We either will die, as everyone does, or we are living "dead" in that we are "fading and withering dismally with age" rather than living life determinedly. Note how the snow falls all over Ireland:
Yes, the newspapers were right: snow was general all over Ireland. It was falling on every part of the dark central plain, on the treeless hills, falling softly upon the Bog of Allen and, farther westward, softly falling into the dark mutinous Shannon waves. It was falling, too, upon every part of the lonely churchyard on the hill where Michael Furey lay buried. It lay thickly drifted on the crooked crosses and headstones, on the spears of the little gate, on the barren thorns. His soul swooned slowly as he heard the snow falling faintly through the universe and faintly falling, like the descent of their last end, upon all the living and the dead.
Snow is presented as a universal phenomenon here, falling upon everyone, "the living and the dead," thus symbolically reinforcing both our eventual fate as humans and also the way that so many of us, like Gabriel, have lived our lives as "dead" people, without really living them.