How does snow connect to the whole story of "The Dead"?
The snow is connected to open spaces and, by extension, freedom in the story. Gabriel yearns to walk in the snow, to be outside, and to be free of the oppressive social situation he faces at the party. He is repeatedly beset by obligation, by pressure, and by a sense of oppression. The snow stands in opposition to this.
"How much more pleasant it would be there than at the supper table!''
Gabriel imagines a dome covered in snow and is pleased by the image. For him, the image is one of simple beauty and even joy.
As the story closes, Gabriel finds that he can engage with the idea of the snow, finally, without being outside. He can commune with the dead - or a sense of the dead - also without ceasing actually to live. Empathy, in his sudden epiphany, frees Gabriel from the constraints of his ego-centric fears.
...he has a sudden realization about his relationship with his wife as well as a realization about himself and the human condition.