The problems posed by the Puritans at the start of the period were focused more on the settlement and changing this; a series of bills in 1571 and 1572 proposed removing the practices regarded as Catholic from the Book of Common Prayer, among other things, but these were all denied. The government dismissed those that put forward the bills, and Church authorities started to penalise Puritan ministers for their more fervent approach. Elizabeth wanted the country to remain united under the settlement, and at this point that still seemed very likely, as the Puritans were not creating a huge furore.
The government imprisoned people such as John Field and Thomas Wilcox who wrote against the practices deemed acceptable by Elizabeth, Puritans only believing in following scripture exactly, and multiple printing-presses were destroyed late in 1572. Still, at this point, the problems did not seem so serious, as it was only minor action that had been taken and no real difficulty faced the government and Elizabeth.
The man mainly behind the Puritan campaign in England was Thomas Cartwright, who was a very extreme Protestant and wrote multiple controversial religious works after being educated at Cambridge. He travelled to Geneva and the Channel Islands, expanding in his Puritanism, becoming more fervent if possible. It was under his leadership that the Puritan complaints moved away from simply focusing on religious matters (which the Queen banned Parliament from discussing unless given permission, thus making it harder for any Puritan politicians to bring up the point) and became more generally about the running of the country and the organisation of the government.
A bill in 1584 was put forward suggesting a change to the Calvinist system operating elsewhere in the world, where there is no monarch. It is called Classical Presbyterianism. This was most definitely the most serious side of the Puritanical movement as it very directly challenged Elizabeth’s own personal beliefs and her role as Queen. John Whitgift, Elizabeth’s third Archbishop of Canterbury during her reign, dealt extremely harshly with this issue through his “24 Questions”, which determined allegiance to Elizabeth’s settlement and safety and eventually led to roughly 400 ministers’ removal from office.
Throughout this time it must be said there were other issues continuing in England. While Elizabeth worried about the threats to the monarch’s authority and the possibility a Puritan-dominated England would be open to Catholic invasion, something she simply couldn’t allow, Elizabeth also had to deal with pressure from the Catholics. The growing tension between England and Spain was hard to ignore, and the loss of trading with the Netherlands, and this meant that they were not deemed to be as great a threat as they could have been as the majority of the rest of the general public turned against them.
In conclusion, I think there was a great variance in the problems caused by the Puritans during Elizabeth’s reign as the objections started out small, much like any other rebel group might, but grew into a direct threat against Elizabeth as monarch. Overall, due to the fact there were the secret Prophesyings and the push for Classical Presbyterianism at one point, the Puritans were a serious problem, but perhaps not as serious as the Catholics.