New England itself was not formerly a colony but a group of colonies that included Massachusetts, Connecticut, New Hampshire, and Rhode Island. The latter three colonies were breakaways from the original Massachusetts Bay colony.
After he was banished from the Massachusetts Bay colony, Roger Williams founded the colony of Rhode Island, which was based on the idea of the separation of church and state. This idea was anathema to the Massachusetts Bay colony's leaders, who believed the church should run the state. Fortunately for Williams, exiled in the middle of winter, the Native Americans helped him survive and sold him land on which to start a new colony. Likewise, John Wheelwright, another dissenter from the Puritans, left to found the New Hampshire colony, while Thomas Hooker established the Connecticut colony. He believed that all adult males with property should be allowed to vote, not only members of the Puritan church.
As you can see, New England grew into several distinct colonies because of religious dissent and different visions of how the relationship between church and state should be organized.