One explanation for why Victorian society believed that Christmas and snowflakes had to go hand in hand is that Prince Albert, the husband of Queen Victoria, was originally from Germany, where snow and ice are common during the winter. As a result, for him, Christmas meant snow. Other well-known Christmas customs were shaped by Prince Albert, including the adoption of the Christmas tree in England, which had previously mostly been a Germanic custom.
The song “I’m dreaming of a White Christmas” by Bing Crosby, whose enormous popularity helped cement the snowflake, snow, and winter chill as being characteristics of iconic Christmas celebrations, seems to be credited, at least in modern times, with helping to cement snow as an integral part of Christmas. The introduction of the snowman and its subsequent unintended adoption is another feature that has aided in this.