The major difference between the Puritans and the Separatists was that the Separatists were more extreme than the Puritans. Some people would say that Separatists were one wing of the Puritan faith while others would say that they were a separate group.
The Puritans were Protestants who believed that the modern church had strayed too far from its biblical roots. They felt that the Roman Catholic Church had been particularly bad in this way, but they also felt that the Church of England (Anglican Church) had retained too many of the characteristics of Catholicism. They wanted to go back to what they saw as a “pure” biblical Christianity. They did not want, for example, the hierarchy and the ceremony of the Anglican Church.
However, the Puritans were not radical enough to want to split away from the Church of England. They wanted to reform it from within. By contrast, the Separatists (as their name implies) did want to form a separate church. They felt that the Church of England was too far gone and that it was not possible to save it.
Thus, Puritans were less radical than Separatists because Separatists wanted to completely break away from the Church of England.
The Puritans were a religious group that practiced the principles of Calvinism. An important difference between Calvinism and many other Protestant sects was the idea of predestination. Calvinists believed that the path to heaven or hell was already determined for individuals before birth. When Henry VIII separated from Roman Catholicism and established the Anglican Church as the national religious institution of England, he retained many of the traditions and rituals of the Latin church. This spawned a religious sect known as the Puritans. The Puritans wanted the English to purify their religion from Roman Catholicism, hence the name Puritan.
A group of Puritans that was more radical in their zeal for Calvinism was the Separatists. The Separatists felt that only those that were predestined for heaven should be allowed to worship in the Puritan church. They pushed for a complete separation from the Church of England. The Pilgrims, who migrated to the New World, were a group of Separatists.