Elizabeth was able to deal with the Puritan threat in Parliament as she had the power to simply close Parliament if they tried to discuss issues she did not want them to. Some Puritans valued good ...
Even if they weren't a great threat, Elizabeth still did not welcome the Puritans' challenge to her authority. 1571 Walter Strickland, leader of the Puritan group in Parliament, wanted to reform ...
So, Puritans and Pilgrims came out of the same religious culture of 1570s England. They diverged in the early 1600s, but wound up 70 years later being one and the same in the New World.