The difference between the Puritans and the Separatists is actually implied in their respective names. The Puritans wanted to purify the Church of England while the Separatists (aka, the Pilgrims) wanted to separate from it entirely.

Both groups felt that the Church of England had strayed too far from the true message of the Bible. They believed that it had been corrupted by worldly ambition and extravagance. Considering that the Church of England was a direct offshoot of the Catholic Church (it had been started in the 1530s over political and personal disagreements between Henry VIII and the Vatican), it seemed natural that the more Calvinist leaning Puritans and Separatists would take issue with how it operated.

The difference between these groups is that the Puritans hoped that their efforts could help to reform the Church of England. They felt that it was still a legitimate religious institution—it had simply strayed. If they practiced their congregational form of religion within the tenets of the official church, they hoped that their English brothers and sisters might come to the realization that the Church of England could be reformed.

The Separatists, on the other hand, felt that there was no hope for the Church of England. They took a more radical approach than the Puritans and set off to start their own sect, entirely distinct from the official state religion. By denying that the King of England and his bishops had any legitimate religious authority, they were terribly persecuted. This is what led many of them to leave England and eventually settle in North America.

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.

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. 

Ironically, the Pilgrim Separatist and Puritan shared belief in the corruption of the Church of England did not lead to a consensus on the best way to institute reforms.

Convinced that the Church of England retained many of its Roman Catholic elements to its detriment, both groups sought change. The Pilgrim Separatists wanted to either break away completely from the Church of England or to utterly destroy the Church. Puritans wanted to stay within the Church to reform it into an image of New Testament purity; Puritans were not Separatists.

The Pilgrim Separatists originally left the Yorkshire village of Scrooby for Holland in 1608. In 1620, they migrated to Plymouth, Massachusetts and started a colony there. The Puritan Non-Separatists followed the trail to the New World and started a colony themselves at Massachusetts Bay in 1630. At the same time, these Puritans called themselves 'Non-Separating Congregationalists.' They eventually became so many in number that their settlements spread to what is present day Connecticut, New Hampshire, Rhode Island, and Maine.

In 1691, due to practical considerations of survival and viability,  the Plymouth colony became incorporated into the Massachusetts Bay Colony. Eventually, dissident groups began to further fragment this temporary 'union.' Out of discord and conflict emerged newly formed groups calling themselves Antinomians, Baptists, Methodists, Quakers, and a whole host of Evangelical Protestant sects well known to present day America.