I go up to Butler regularly, including in the winter. Were you thinking of parking at stevensville rd on the underhill side? Or were you looking at the stowe side (I know side that much less well, but I have parked on the road to ski). The maps that show the ski trails are usually marked as the ski and snowshoe maps, and just to make things more fun, the names of some of the trails have changed over time. In any case, there are a ton of different trail options in those areas, all are going to get quite a bit of use.
Taylor is newer and fancy and nice, with I would bet a nicer view; Butler is smaller and more rustic. Taylor is lower and not really near a summit. Either would be fine for an unheated cabin overnight, in which, it can obviously get very, very cold.
The skiers hit the ski trails pretty regularly, but you can snowshoe on them, just don't posthole the uptrack. Nebraska notch is skied and snowshoed all the time. The Butler lodge hiking trail is also likely to be packed down most all the time up to the lodge and is easy as pie. Depending on recent snow, you may use microspikes all the way; sometimes near the top there is a lot less traffic and lot more snow and shoes come in handy. Hard to predict Dec, but there will likely be snow, several feet deep.
Once you get off the main trunk trails going up though (butler on shoes, underhill/overland on skis), all bets are off. I have had a difficult time finding the long trail in sections along that ridge, and have broken trail from Butler to Taft, which you would think would be popular. Route finding on the ridges can be difficult and the going can be very challenging, drifts, spruce traps, ice, etc. I have thought about bushwacking Mt Clark, but never Dewey. I would guess it has been done, it could be difficult and alone, maybe far too much work.
Even the long trail north of Butler to the summit can be impossible to find and impassable; it may be open in Dec still, but I have given up thinking I can break that trail myself when I have tried it. Most/all traffic out of butler seem to take Wampahoofus to Maple Ridge instead, that gets steep and dicey in parts and I haven't actually been all the way up in mid winter, but I am sure is doable and a great, gorgeous hike with moderate exposure.
Some of the ski trails on the map are never used and essentially lost and you would be breaking trail the whole way.
Enjoy your trip to VT.
edited to correct that I was looking at Taylor lodge, not Taft. Taylor is in Nebraska Notch; Taft is under the summit of the Chin.