They were called Pilgrims because, as English Separatists, they wanted to separate from the Church of England and establish religious freedom for themselves in the New World. As they traveled to form their own religious congregation, they were called Pilgrims.

Note that though the English Separatists held Calvinist beliefs like the Puritans, the Puritans actually did not want to separate from the Church of England, but rather "purify" it by injecting their own theology. So Puritans and Pilgrims are not the same thing.