The dirty, slushy snow in "American History" by Judith Ortiz Cofer symbolizes urban depression. Think of snow as one of the stages of the never-ending water cycle. It is the form of precipitation that seems to "stick around" the longest and the most. It is also the state of water which causes the most happiness (when it is white and fluffy) and the most issues and annoyance when it is dirty and slushy.
Every Sunday we drove out to the suburbs of Paterson, Clifton, and Passaic, out to where people mowed grass on Sundays in the summer, and where children made snowmen in the winter from pure white snow, not like the gray slush of Paterson which seemed to fall from the sky in that hue.
The implication is that, from where Ortiz stood, not just geographically, but also emotionally, everything that could happen to her would have to be gray, and ugly, and malformed. Nature had a pre-disposition to send her the ugly stuff.
It is no wonder; the story is about personal and universal tragedy and misery. She compares the thoughts and dreams of her parents, the talks about the beautiful beach weather of Puerto Rico, with her current situation: with the cold, long winters, and with the dirty gray slushy snow that comes down just her way. In her view, the much luckier, brighter, happier suburban kids get the white fluffy snow. It is all a metaphor to compare the situations and how she felt inside and out.