You have identified an important symbol in this excellent short story. The snow that dominates this tale is something that is shown to link both the living and the dead in the state of paralysis that dominates all of the characters, save for the active and vibrant Michael Furey. If you look in particular at the way that the snow is presented in the last paragraph of the story, it is clear that what is emphasised is the way that the snow unites the dead and the living together, even Michael Furey himself. As such, the snow is shown to be universal:
Yes, the newspapers were right: snow was general all over Ireland. It was falling on every part of the dark central plain, on the treeeless hills, falling softly upon the Bog of Allen and, farther westward, softly falling into the dark mutinous Shannon waves. It was falling, to, upon every part of the lonely churchyard on the hill where Michael Furey lay buried... 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.
Note the universal reference to the "universe" and the snow falling "upon all the living and the dead." The snow therefore symbolises the paralysis that is demonstrated by Gabriel Conroy throughout the short story. He is constantly concerned by what others think about him, and this stops him from truly living. This is what he realises in his epiphany before the snow starts again. The snow emphasises the way in which "living" and "dead" are somewhat blurred categories. There are some people who are living, like Gabriel, that, through their paralysis, have never actually "lived," whereas Michael Furey is an example of a "dead" person who has lived far more than Gabriel ever has, in spite of his early demise. Yet the snow unites them both, mocking Gabriel's earlier thought in his speech that we must not brood over the dead. Gretta cannot do this, and at the end of the story, Gabriel realises how futile such attempts are as well. The snow could also symbolise his own emotional isolation and coldness, but remember that this is a condition that is applied all across Ireland.