There are puritans and puritans.
The understanding of the word puritan has evolved. The name `puritan' was given to those in the Church of England, reformed by Elizabeth the First who didn't think the reforms went far enough. These tended to be people who held more to Scripture, prayer and sermons, whereas the mainstream was more ceremonial and seen by the puritans as `papist'. The modern perception of a puritan as a nasty old prurient busybody who likes stopping people having fun, derived from the more authoritarian expressions of the puritan faction.
The Elizabethan church produced heresies of its own, the most interesting being English Anabaptism. This is not to be confused with the more radical Dutch or German Anabaptism that produced some quite fascinating attitudes such as Shakers and the like. The English variety sat very well with the surviving elements of Lollardy and in due time fragmented into both General Baptism and Particular Baptism, each of which in turn evolved in different ways. However, these are the originators of the Baptist churches of today so have, generally speaking, developed a modern, tolerant style.
The New England colonies were the product of particular congregations which would only accept new colonists on the basis that the newcomer accepted their brand of belief. They were not generally tolerant and so tended to fragment over time. It is the fragmentation that produced division and hysterical accusations of heresy and witchcraft. The creation of Rhode Island is a useful illustration of the early process of fragmentation.
It is usual these days to see New England as the core of the incipient United States. In terms of individualism I agree. The can-do Yankee attitude of the pioneer has its roots in the puritan ideal of Scripture and prayer. Yet, the original colonies also included plantations that employed slave labour. Such estates could provide a much quicker return on investment than any settlement of peasant farmers, however much they cooperated with each other. I tend to view the slave plantation built on land stolen from Native Americans is a truer statement of how the United States began than peasant farmers in New England ploughing enclosures with the bible in hand and cooperating with their neighbour next door.
My own cultural origins are in puritanism. The English side were Anabaptists in the 17th century and the Scottish side were largely for the Covenant. My background covers both dour Calvinism and lively Quaker argument. I have to own up to both despite an Anglican upbringing. The puritans are difficult people to understand as to us conscience is everything.