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Both the Pilgrims and the Puritans had religious roots based on Calvinism.
The Pilgrims wanted to move and establish their own community, operative word being their own). They wished to isolate themselves and live in a way that they could practice their religious beliefs a part from persecution and outside influence. They began by moving from England to Holland, and then moved to America.
The Puritans originally had no desire to leave England, but rather wished to stay and purge the country of its Roman Catholic influences. A group of Puritans left England and settled in Massachusetts Bay because of increasing persecution during the realm of Charles I. Many stayed in England.