Both Puritans and Pilgrims came to the New World with the means of practicing their own religion. However, these two groups differed on how what religion they would practice.
Puritans wanted to continue to associate themselves with the Church of England (aka the Anglican Church). They wanted to "purify" this church and focused on fixing the mistakes made by the Anglican Church during the 15th and 16th century.
On the other hand, Pilgrims wanted to break away completely from the Church of England. They moved to the New World in hopes of creating a new religion that was less corrupt than the one they left in England.
