Frozen rain, or melty snow, is called sleet. Walking in the snow can be lovely, but walking in sleet is just cold and unpleasant.
Sleet is a type of precipitation, and it's also a verb: "Is it starting to sleet out there? Let's stay in and watch TV tonight." In the U.S., sleet is the name for small ice pellets—if you've ever heard the weather person talk about a "wintry mix," it usually includes sleet along with snow and freezing rain. The word dates from the very early 14th century, from a Germanic root.