How can we explain the fact that Pilgrims and Puritans escaped persecution only to come to the New World and pass judgement on others?

We can explain this apparent paradox by simply observing that, for the most part, neither Separatist Pilgrims nor Puritans shared our modern concept of religious freedom. What they were after when they left England to settle in North America was the ability to create settlements which would become, in the words of the Puritan founders of the Massachusetts Bay Colony, a "bible commonwealth." They wanted to create societies that conformed to their vision. A big part of this vision was community. They thought that people who did not share their beliefs were a threat to this community, and they thus dealt with these people quite harshly. For example, when Roger Williams and Anne Hutchinson publicly differed with the leadership of the colony, they were banned to Rhode Island. Many people, such as Quakers, were actually executed.

Puritans believed that the colony would not survive if people they viewed as ungodly were allowed to remain. This belief would be tested, ironically, as a result of the colony's success. It spread so fast and witnessed such rapid economic growth that the sense of community the founders of Puritan New England colonies sought to create came under threat even before the founders themselves had died.