The Dedication of Our Lady of the Snows has simply become the Dedication of the Basilica of Our Lady. Here's the 3rd reading from Matins in the EF:
When Liberius was Supreme Pontiff, a Roman Patrician called John and his wife, also of noble birth, who had no children to inherit their goods, vowed to bequeath everything to the most holy Mother of God. The blessed Virgin hearing their prayers and vow, approved them by a miracle. On the Nones (fifth) of August, when the heat in Rome is at its highest, there was during the night a fall of snow on a certain part of the Esquiline hill. And on the same night, the Mother of God spoke separately to John and his wife in a dream that they should build a church on the same spot. John told this to Pope Liberius, who confirmed that he had received the same dream. So they went to the snow-covered hill and there laid out a site for the new church, which John and his wife erected at their own cost, and which was later restored by Sixtus III. It has been called by various names, but so that its title may signify its excellence, it is best known as the Basilica of St Mary Major.
Well, I find nothing to strain credulity in this story; meteorological freaks have occurred often in improbable places. Andrew, my Sydney correspondent, tells me they had a freak snowfall there (soon to be posted) recently; the first in a couple of hundred years. I sadly saw this morning that even our new Martyrology makes no reference to the legend.
It does, however, mention that today is also the feast of Saint Oswald, converted King of Northumbria, who died in battle against the pagan Mercians at Oswestry. More here.