I’m reading a book called Tiw by William Barnes, where it’s suggested the words wormwood & vermouth come from ‘ward’ + ‘moth’. I noticed the entry for wormwood on the Online Etymology Dictionary says the ultimate etymology is unknown, so I thought I’d copy what Barnes said here, in case it’s helpful.
It would appear from the spelling that “vermouth” in English might be a borrowing of a French calque of German Wermut(h), or something close to that. I think your two words could be roughly considered “the same word for the same thing, more or less, -ish”. I never knew of the possible moth connection - but apparently the word moth in older Germanic languages also meant “worm”, so maybe all the English and German(ic) forms of this word are literally talking about the same thing; the English “-wood” ending could be an attempt at translation, or just a folk etymology/eggcorn. (“Ver” in French even happens to mean “worm”, though I guess that would be an accident.)