I should probably lump the Separatists in with the Puritans, but decided to give them their due in a "separate" post. The Separatists wanted a complete break from the Church of England, while the Puritans wanted to make a few changes.
One of the primary Separatist leaders was William Brewster of Scrooby, Nottinghamshire, England. The Separatists were tolerated in England, but subject to investigation and criticism. In 1607, Brewster decided to follow a group from Gainsborough to Leyden, Holland, where the dissenters were welcomed. Ten years later, the Separatists decided a new home was required. Their children were adapting to Dutch customs and culture and financial opportunities were starting to wane.
Brewster, Bradford and the other leaders elected to sail for the recently established Virginia colony. The Mayflower was blown off course and arrived far to the north, off the coast of present-day Massachusetts in 1620. Barely surviving the first winter, about half of the Separatists and other passengers established the Plymouth Colony.
Brewster remained one of the leaders of the new colony. Elder Brewster was joined by his eldest son Jonathan in 1621. Jonathan established himself as a ferryman and Indian trader.
John Oldham, Jonathan Brewster's brother-in-law, was a coastal trader and some of his ventures were a bit shady. John met his end on the coast of either Block or Fisher's Island in 1636, when Indians attacked his boat. This was one of the events that ignited the Pequot War.
The Separatists, for the most part, eventually became members of the Congregational Church founded by the Puritan settlers.