In contrast to the pilgrims, the puritans wanted to:

Purify the Church--Pilgrims were separatists, wanting a new religion but Puritans wanted to fix the problems and make the Church more moral. 

Puritans aimed at creating a moral church without the glam of Catholicism. Catholics had become flashy with colored clothing, dancing, drinking, and even priests having children. Puritans focused on hard work and a simple life to prove worthiness for heaven.