“There is no hard line between Separatists (i.e. the Pilgrims) and Puritans.”
Except they separated from the Puritans, ie, they no longer considered themselves Puritans.
The Pilgrims founded the Plimouth Colony, the Puritans the Mass Bay and Salem Colonies.
No one categorized as a "Puritan" called himself a Puritan.
The Pilgrims were Separatists or Nonconformists, the Massachusetts Bay colonists were Conformists.
Both groups held to views of Scripture that were characterized by mainstream Anglicans as "Puritan."
Both groups also sided with the Parliamentarians in the Civil War and sent soldiers to fight on behalf of the Parliamentary forces - this can be seen as a classic litmus test of Puritan sympathies.
It was Browne's student Richard Clyfton who remained separated when Browne did not and began a congregation in William Brewster's house.
After Clyfton died, Brewster became the senior elder/pastor of the congregation that would eventually become the congregation of the Plymouth colony.