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First, I don't think that there's a link, for example in the meaning, Between these two names. I reckon your husband's grandfather dropped a difficult name and adopted one that was easier for the host population to get along with, i.e., for reading, spelling and pronouncing. It's likely he chose a family name, say his mother's maiden name, or a version of it.
I think Budwetski is Polish, though in an area of shifting borders, which might have been classed as Russian at some stage in its history. I think it would be spelt Budwecki in the Polish language, the letter C being pronounced -ts-, even when followed by the letter K. It's probably from a place name, maybe Budwiec, near Suwalki in North-Eastern Poland.
There is website that lists the Jewish surnames of Suwalki,and the surnames Budwecki and Budwiedzki are among them -
http://www.jewishgen.org/JRI-Pl/psa/suwalkisurn.htm
LaVine I take to be an alternative spelling of Levin, from the given name Levi.