The Pilgrims and Puritans were not the same. They both had their roots in dissension from the Church of England, which became known as “Puritanism.” In America we use a narrower meaning for Puritan.
We refer to the Puritans as those who came in Winthrop’s fleet of 17 ships in 1630. They settled the Massachusetts Bay Colony. In the beginning they were loyal to the Church of England. They were rigid in their theology and arrogant towards their neighbors.
Members of the Separatist group who came on the Mayflower in 1620 are referred to as Pilgrims and formed the Plymouth Colony. Not all on the Mayflower were there for religious reasons. Some were just other emigrants or those hired by the Pilgrims. They all eventually became known as the Pilgrim Fathers. They were more tolerant, less judgmental and more fair in their dealings with native Americans.
*Early in 17th century England it was a crime not to attend the National Church. Those not attending were punished and the church kept lists. Perhaps some of your ancestors are on those lists.