The issue is that rural white NE, especially Vermont, is not as evangelical or ultra religious as other rural white areas of the US.
This is the key I think. New England, both in rural and urban areas, is quite secular--and religious people are likely to be Catholic, and if not Catholic, mainline Protestant. New England is the least evangelical part of the country, and that applies to rural areas too.
In any case, a lot of New England is actually not that rural--virtually all of Connecticut and Rhode Island and most of Eastern Massachusetts and Southeastern New Hampshire are functionally urban or suburban. And of course you have small cities as well, like Portland or Burlington, outside of those larger cities. These places voting Democratic shouldn't be a surprise.
Outside of these areas, you sort of have to take it region by region. The Berkshires and Pioneer Valley are a dense network of postindustrial towns and also a lot of them have prestigious colleges--so unsurprisingly they have a certain rural hippie element, a bit like Albemarle County or the rurals around Madison, on top of an already vigorous union presence. And of course Western Mass is near both NYC and Boston in the grand scheme of things so if you're a left-leaning person who want to live a rural lifestyle it's an obvious location to move to.
Vermont is a bit similar, except with a weaker union element but IIRC an even stronger "back to the land" movement. There are similar dynamics at play in western New Hampshire. In both NH and VT, there are rural areas with a weaker college/hippy element, and those places tend to be closer--but usually still Dem-leaning due to the SECULAR/Yankee factor, at least in less remote areas.
Cape Cod and the Islands are heavily dominated by vacationers from major cities or inner suburbs, who are generally Dem leaning these days. Coastal Maine is similar too, plus some colleges, etc, and IIRC it was also somewhat industrial in parts.
That said, there has been some growing erosion in certain rural areas of New England. Trump gained quite a bit in traditionally Democratic ex-industrial areas in Northern Maine and New Hampshire, and Republicans have never quite lost strength in some especially remote regions, like the Northeast Kingdom, the Lakes District of New Hampshire, and rural northern Maine.
