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.