All Puritans were not the same. While all of them feared that King Charles would never allow the reforms they wanted, not all of them wanted to break from the Church of England. The Puritans who became known as "Separatists" eventually settled in Plymouth; the Puritans who believed it was important to remain connected to the Church despite its failings settled the Massachusetts Bay Colony.
The Puritans in Plymouth signed the Mayflower Compact before they ever landed in America; this became the governing document for the colony. The signers,
in the presence of God, and one of another, covenant and combine our selves together into a civil body politic, for our better ordering and preservation and furtherance of the ends aforesaid; and by virtue hereof to enact, constitute, and frame such just and equal laws, ordinances, acts, constitutions and offices, from time to time, as shall be thought most meet and convenient for the general good of the Colony, unto which we promise all due submission and obedience.
This agreement bound the signers to obey any laws, present and future, they deemed necessary for the Glory of God and the public good. Their laws were based on Biblical law and the common good of the colony, and compliance was based on their solemn vows. Because they were like-minded in their faith and purpose, the first generation of Plymouth Puritans experienced little dissension. As others, perhaps not so like-minded or godly, joined the group, Plymouth began to have trouble.
The Massachusetts Bay Colony, though founded by Puritans, was as much an economic venture as a religious pilgrimage. Traditionally, to be a voting stockholder in a trading company such as this, one had to be a landowner; in order to be a stockholder in the Massachusetts Bay Company, one had to be a member of the church.
Because these Puritans were Calvinists (they believed in God's elected or chosen people), they felt a great responsibility to ensure--by the laws they created and by the strict enforcement of them--that God's people remained pure so they would be ready for Heaven if they were among the Elect. John Winthrop described it this way in a sermon:
[W]e shall be a city upon a hill. The eyes of all people are upon us, so that if we shall deal falsely with our God in this work we have undertaken, and so cause Him to withdraw His present help from us, we shall be made a story and a byword through the world.