The Puritans were not happy with either Elizabeth of James; however they did not constitute a serious threat to either. This was primarily because they had no established theory of religion; they only knew that which they were against.
Elizabeth I had been subjected to merciless abuse and barely escaped execution herself at the hands of her half sister, Mary I, who was determined to return England to Catholicism. Elizabeth, who was quite pragmatic, developed a religious settlement which preserved all the pageantry and ritual of the Catholic service, but allowed some degree of Protestant belief. This was not enough for the Puritans, who wanted to "purify" the church from any element of "popery." It is from this that the name "puritan" originated. The term was pejorative; they called themselves "the Godly." The Puritans criticized Elizabeth heavily for not eliminating the vestments of Catholicism, but had no direct influence on her religious policy. They were more of an annoyance than anything else.
James I, formerly James VI of Scotland, had thirty five years experience as ruler of Scotland, and should have been as well versed as Elizabeth in dealing with religious dissidents; but was too proud of himself to be bothered. He also considered himself to be something of an armchair theologian. He was called the "wisest fool in Christendom." James was Calvinist, but not a Puritan by any stretch. At one point, the Puritans insisted that he abolish the office of Bishop in the Church as this constituted popery. His reply was "no bishop, no King." Although the Puritans gained seats in the House of Commons, they again constituted o threat to James. His son, Charles I was not so lucky. It was a Puritan House of Commons under Oliver Cromwell that sentenced him to death.