That is a complicated answer. By pilgrims, do you mean those who landed in Massachusetts in the 17th century? Because those were puritans who were acting as pilgrims who landed there.
A pilgrim is a person from any faith who goes on a journey for religious reasons. The puritans, in moving from England to what is today Massachusetts, were pilgrims, moving due to religious persecution. They were called puritans because they believed in a fundamentalist reading of the bible, believing it to be 'pure.'