But you must separate one from the other. During the latter half of the 16th Century and the beginning of the 17th there were several movements in England to reform the Church of England. But Queen Elizabeth I, King James I and King Charles I, as head of the church, would have none of it. And as time wore on, the Kings moved to prosecute and jail those who disagreed with the dictates of the Church of England.
Both the Puritan group and what we know as the "Pilgrims" believed the Church of England to be too "Papist." They felt the Church of Rome was still extending its influence. And this is where the Puritans take their name, the desired to "purify" the church from any and all of its Papist tendencies. They desired to do this from within the Church, meaning, they would do their best to remain in good standing while trying to effect changes.
The "Pilgrims" are more correctly called "separatists." It was their belief that the Church was incapable of change and it was their desire to separate into an entirely new religion. But such a move, in those days, was not legal according to the reigning monarch. Their leader, the outspoken John Brewster, had a price on his head and all his followers were subject to being jailed.
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