When the Pilgrims lived in England, they were called Separatists and considered an extremist group. Dissatisfied with the lax rules of the Church of England at the time, the Separatists formed an illegal congregation apart from Queen Elizabeth I's Church of England.
Some of the Pilgrims were known as Puritans when they were still in England. They were a less extreme version of the Separatists. The Puritans believed that the Church of England needed reform in certain areas, but they didn't withdraw from it completely. Many Puritans eventually joined the Separatists in their journey to America, where they formed a farming colony in present-day Virginia.