The Puritans didn't think that the Church of England was sufficiently Protestant in its doctrines and practices. Puritans were Calvinists and looked to Calvin's Geneva as the model for the kind of theocracy they wished to see established in England. But the Church of England, especially after the Elizabethan settlement of 1559, was always something of a compromise between different factions. This was thought to be the only way to maintain peace in a kingdom fractured for decades by bitter religious conflict.
But the Puritans weren't satisfied with this compromise; they believed that the Elizabethan settlement retained too many of the old Catholic elements. In particular, some of the more radical elements among the Puritans railed against an episcopal form of church government, that is to say a Church formed on a hierarchy of bishops. Many Puritans wanted a much simpler, less hierarchical form of church government, with much less visible pomp. However, Queen Elizabeth I fiercely resisted any such proposals. As far as she was concerned, an episcopal form of government was an essential part of a society ruled from the top down. If the bishops could be got rid of, then the same principle behind Puritan church reform could just as easily be applied to the social order as a whole, undermining the power of society's traditional rulers.