According to this article, there were multiple factors involved but the conclusion appears to be that the deciding factor was migration:
Vermont and New Hampshire, Geographic twins, Cultural aliens
Here is a very intriguing intro, which leads one to believe what @bike says.
There is no simple answer, but, he said, there were many wrong turns in a search for illuminating data, which had to span at least 50 to 100 years.
Can demographics explain why the two states diverged? Nope. A century of census data on Vermont and New Hampshire shows that both states remain “extremely white” and largely Christian, said Kaufman.
How about economics? Another false lead, Kaufman said. Vermont is somewhat more agricultural, though not enough so to explain the cultural divergence. And both states have a small manufacturing base.
Educational profile? “Really no difference,” he said.
Here is the article's conclusion.
In sum, Vermont engineered a new brand for itself — a hybrid culture of the old and the new. New Hampshire held on to a brand that was a variation of the old one.
All this took “an unintentional concatenation of forces,” said Kaufman. “[But] Matt and I think the migration factor is the key.”