It's easy to confuse the Pilgrims and the Puritans, especially since they both wore funny clothes. Not only do both titles begin with P, both groups left England to achieve Protestant religious freedom in the New World, and both groups shared basically the same theology and many of the same ideals and social theories.
The biggest difference was that the Puritans wanted to cleanse the Church of England from pernicious doctrine and practice, while the Pilgrims believed it had already grown too corrupt for rescue and made a break with the institutionalized Church. The Pilgrims were a lot like the children of Israel, led out of the richness of Egypt to an inhospitable wilderness in the name of Yahweh.
Ultimately, the Pilgrims and the Puritans weren't much different. The ideas the Pilgrims brought with them, first to the Netherlands and then to the newly settled American continent, were the ideals that drove the Puritans: personal piety, a devotion to and joy in God above all else, Reformed theology, and the Church as a spiritual entity separate from the secular State.
These are the people who landed at Plymouth Rock in 1620. They were twice exiled from their homeland, but instead of surrendering to despair and discouragement, these hardships strengthened the Pilgrims' resolve and commitment to their convictions. Glory was attributed to God alone, as demonstrated most notably in the first Thanksgiving celebration which followed their arrival in what would become Massachusetts.
Everyone knows about that First Thanksgiving, how the tables were heaped with corn and turkey and venison and beer, how the Native American guests entertained their European hosts, how the whole enterprise was God-centered and joyous. What's not so well-remembered is the fact that the Pilgrims were in fact thanking God for safe passage from the Old World, a "safe passage" that whittled their number from 102 to 53 through disease.
Paul speaks of the joy of suffering in Christ's name. We don't often really understand what that means, but the Pilgrims exemplify this attitude. They had come through adversity of every kind (persecution, exile, starvation, disease) in their pursuit of Jesus Christ and their dedication to the Gospel, and even in the midst of these hardships, they gave thanks. We humbly hope and pray we will do the same in whatever circumstances we find ourselves.