The Pilgrims came to what is now the United States for the opportunity to practice their own religion, while the Puritans came to reform the beliefs of the Church of England and then spread them. The puritanical view was much more restorative than that of the Pilgrims.
The Pilgrims first arrived in what is now Massachusetts in 1620. They fled from England after being persecuted for their beliefs. They first sought exile in the Netherlands because Dutch laws were lenient and allowed them to practice their religious ideals freely. They became concerned about the loss of the English language and culture and 100 pilgrims eventually decided to journey to North America. They did not seek religious reform, only the freedom to practice their religion within the boundaries of their own culture.
The Puritans, in contrast, were British colonists who vehemently disagreed with the role of the king as the head of the church. They believed that the tenets of the Church of England were too closely related to Catholicism. The Puritans hoped to begin a period of Protestant restoration in the "new land" similar to that taking place in Europe. Their goal was to reform the practices of the Church of England and then convert followers.