Pilgrims and Puritans are not interchangeable terms, nor is one a subset of another. They shared a common faith, the Anglican Church (aka Church of England) and, as colonial neighbors, often collaborated. Both believed in purifying and simplifying Anglican practices and that Scripture was the guiding source for a community’s moral codes. Both groups also advocated literacy, so that each person could read the Bible.
Pilgrims settled Plymouth Colony beginning in 1620 with the arrival of the Mayflower. They continued to settle Plymouth towns into the 1630s.
Puritans were the founding families of Massachusetts Bay Colony (Arriving 1630 onward, Boston, western Massachusetts and into Connecticut.).
Pilgrims insisted state and church should be separate, and were known as Separatists. To be clear, as an astute reader pointed out, the Pilgrims earned the name Separatists because they had dissociated with the church of England — not simply over their stance that church and state not bleed into one another’s activities. At this point, we can’t go much further without clarifying a few things. Not all passengers on the Mayflower were religious dissidents. The passengers who left England to separate from the Church of England referred to themselves as Saints. The Saints referred to the other passengers as Strangers. For our purposes here, we’re referring to both groups as Pilgrims.