Superficially...
Quakers are a peace-loving movement who tend to keep to themselves as they practise their fundamentalist faith. They're generally pretty nice people.
Pilgrims (in general terms) are those people on a spiritual journey (like the Christian version of the Islamic Haj). As a sect, they were a break-away from the Separatists (who were themselves breaking away from the Church of England). From an American point of view they were among the early settlers of Massachusetts.
Puritans? Easy peasy. They're a group of narrow-minded, self-satisfied, holier-than-thou, pains in the butt. Puritans believe anything that gives pleasure is "sin". As one poet put it, "The puritan through life's sweet garden goes, to pluck the thorn and throw away the rose."