The Puritans gained most of their power following the first English Civil War, and most of the Puritan ministers renounced the Church of England following the English Restoration of 1660 and the subsequent Uniformity Act of 1662. Puritans believed that the Church of England still maintained many characteristics similar to the Roman Catholic Church, and they supported a greater form of purity of doctrinal worship. Highly anti-Catholic, the Puritans believed that the Church of England required further reform. Additionally, they opposed the idea that the king should be the supreme ruler over the church; instead, they believed that only Christ could rule the church--be it in heaven or on earth. Puritans believed in a minimum of ritual (no use of candles or artistic images) and decried excessive preaching; like the Calvinists, Puritans also supported a strict regulation of worship and were anti-traditionalist. Puritans did not support or celebrate traditional religious holidays.
The Puritans were infuriated by the marriage of Henrietta-Marie de Bourbon to King Charles I, King James' son and successor, in 1625. She was a Roman Catholic and decidedly anti-Puritan. The Puritans also despised King Charles' advisor, William Laud, who also disapproved of the rise in Puritanism power. Charles later used his Star Chamber and Court of High Commission to suppress Puritans by conviction and imprisonment.
The basic problem that the Puritans had with the Church of England was that it was, in their minds, too much like the Catholic Church. The Puritans thought that the Church of England had not done enough to purify itself of Catholic influences.
Two specific disagreements were over church hierarchy and the nature of the worship service. The Puritans did not believe in a church hierarchy with bishops and archbishops and such. They believed that each congregation should be autonomous. The Church of England was, to them, too hierarchical. The Puritans also believed in simple church services centered around a sermon. They belived the Church of England's services were like Catholic masses and therefore too ritualistic.
As these examples show, the Puritans' major problem with the Church of England was that it was too much like the Roman Catholic Church.
In the Puritan discipline, dancing was acceptable, but sexual dancing was not. Drinking alcohol was also acceptable but becoming a drunkard was not. The Puritans believed very strongly in marriage and were opposed to illicit sexual activities. Adultery was punishable by death, and fornication was to be punished be whipping. They wanted to "purify" the Church of England and put an end to the hierarchy that led to corruption. They believed that the church should follow the scriptures exactly. There was a dislike of the Pope's practice of selling dulgences and the massive ornamentation of the Church.
The puritans believed in self-determination, that each has the ability to do good. That view exists today. They also stressed education and the desire to reform by education and good works are the heart of American democracy.
If you return to the original Lutheran Reformation of the Roman Catholic Church, you find that implicit in Martin Luther's original articles of faith--the priesthood of all believers, the authority of the Scriptures, and the doctrine of salvation by faith alone--is the rejection of the ecclesiastical hierarchy (and bureaucracy) that had evolved since Christ gave the keys to the Kingdom of Heaven to Saint Peter. Luther's famous rejection of Pope Leo's indulgences led inevitably to an outright denial that the Roman Catholic practice of Christianity was still consistent with the original precepts of Christian faith. Of course, because he had to maintain good relations with Frederick, Elector of Saxony, for his very survival, there was no way Luther could continue that line of thought to its logical conclusion that a priesthood of believers meant a community of equal brothers, which would certainly threaten the social and political order of the day.
However, in England, where the Reformation had been rather cynically imported by Henry VIII to justify himself as Head of Church and State and to relegate the Pope to no more than the Bishop of Rome, many churchmen did pursue Luther's original train of thought and called into question whether aristocracy was essentially incompatible with Christian practice in its purest form. (This was not a new development, by the way. John Ball had raised the same questions in the reign of Richard II, preaching in support of the Peasants' Revolt against the very notion of the upper and lower classes. Jan Hus had made a similar, although less radical, protest in Bohemia in the early 15th century. Needless to say, those ideas were easy to crush when they were merely local; with the printing press, however, ideas took flight and the call for reform was everywhere.) Puritanism demanded that the English (Anglican) Catholic Church extend its reforms of Roman Catholicism and return to the simple community of brethren and believers that they read about in the description of the early Christian Church in the Book of Acts.
Needless to say, not only did the Puritans have a problem with the Church of England, but the Anglican hierarchy--which was in most respects quite similar in practice and structure to the Roman Church--had a fundamental problem with the Puritan vision of reformation. When Henry was enforcing his will on both church and state, he famously sent both Roman Catholic and Puritan clergymen to the stake in pairs, chained to the same pyre, for the offense of not subscribing to the middle ground of reform that he dictated for his people.