At first sight, a connection between Rain and Snow and The Holly Twig seems tenuous, but you never know. Perhaps there are "missing links" that would make it more clear. The latter song, according to Sabine Baring-Gould, appeared on English broadsides in the later part of the 18th century. You can see a C19 edition at Bodleian Library Broadside Ballads:

Weeks work completed.

Of course, there have been plenty of other songs on the same basic subject.