Both the Separatists and the Puritans opposed Anglican Church practices which they considered unbiblical (especially practices which had been brought over from the Catholic Church). Additionally, both groups faced persecution from Anglican authorities for their opposition to the official state church.
However, the Separatists and Puritans responded to these practices and this persecution in different ways. The Separatists, as their name suggests, believed the Anglican Church to be so corrupt that it could not be redeemed. Consequently, they left England for the Netherlands and eventually the New World (settling in Plymouth in 1620). The Puritans, as their name likewise indicates, believed they could purify the Anglican Church and restore it to what they considered a biblical model of church practice and government. However, the Puritans found the political and religious situation in England to be increasingly hostile, so they began to leave England as well. From 1630-1640, around 80,000 Puritans settled in the New World during the Great Puritan Migration, with most settling in Massachusetts.