The Puritans were Anglicans trying to purify the church of England, which is why they were called “Puritans”. The Pilgrims were separatist (independent of the C of E) congregationalists. In time, the Pilgrims led the New England Puritans to become “Congrgationalists.” But, originally, the Puritans were Anglican. BTW, at the time the Anglicans had a reformed confession of faith. Consequently, most of the difference between Episcopalians/Anglicans and Congregationalists was in the area of ecclesiology.
Overall, the reformed Christians in New England - many of them Puritans - had more influence culturally in the formation of the country than Anglicans. The lines of influence were somewhat blurred, however, because they shared so much theology.
The cultural heritage of America and particularly the religious freedom attained by disestablishing State Churches is rightly attributed to the many Protestant groups persecuted by Puritans, Anglicans and to some extent Congregationalists. Scotch-Irish Presbyterians, Baptists, Lutherans, Dutch Reformed, Moravians and others.
Referring historically to the Puritan groups, the first of which accidentally blundered into Cape Cod and landed at Plymouth due to a navigational error while on their way to their land patent from, yes, The London Company in Viriginia, as the founding group of this people and this nation is nothing but sanitized myth, popularized during and after the Civil War to strip the southern states of their role in the nation’s heritage.
For every so-called intellectual, moral or religious heir of New England Puritans, there are two or more of Virginia and her descendants in other States. The Bill Of Rights, guaranteeing religious freedom, is modelled after that of Virginia, penned by Founder George Mason. Mason was tremendously influential in his era, regarded as perhaps the most necessary man among our Founders, the intellectual equal or better of any one of them.
Mason risked his personal reputation and lost friends such as fellow Virginian and neighbor George Washington, over his refusal to put his hand to the Constitution, precisely because it provided no bill of rights. He did not sign, he walked out. James Madison, Virginian as well, attempted to heal the rift, and the very first Amendment was immediately proposed in deference to Mason.
Not a New England Puritan in the lot. Some New England States retained their State Church into the 1800’s. Southern ones including Virginia largely disestablished theirs prior to ratification of the Constitution, under the Articles Of Confederation.