It was ALL about religion
In 1605 in England, there was no religious freedom. The Church of England was the state religion, and an attack on it was an attack on England. Although this church was the result of the Protestant Reformation, there were individuals who felt that the church still retained too much of the liturgy, sacraments, and ritual and had not gone far enough to reform the church. They referred to themselves as "Separatists."
They had to meet in secret and not promote their views among the rest of the population. Doing so could result in a loss of livelihood, imprisonment, and even death. Conditions became so difficult that a group of about 80 Separatists decided to relocate to Amsterdam and, later, the town of Leiden. At the time, it was illegal for Catholics and Protestant Separatists to emigrate without permission. In 1607, part of their group was able to escape to Amsterdam before being discovered. The rest of the group was able to join them in 1608. William Brewster of Mayflower fame, became one of their leaders after losing his job as postmaster in Scrooby, England because of his religious beliefs.
The reality of life in Leiden, Netherlands did not live up to their expectations. As they were not citizens, they were limited in the kinds of employment they could obtain.Their children wanted to adopt the ways of the Dutch. And so the decision was made to reestablish a community in the New World.
On September 5th, 1620, 102 passengers and crew, departed for what turned out to be a 66-day passage. The voyage started out smoothly, but by October, they encountered fierce Atlantic storms that made the voyage treacherous. The winds blew them off-course, but they finally landed in what is now Cape Cod on November 9th. This late fall arrival date forced the passengers to rely on the provisions on board, and most of the winter was spent living on the boat. Food dwindled, disease spread, and temperatures dropped resulting in the deaths of half of the passengers and crew that first winter.
I could write a book about the rough beginning in what came to be known as Plymouth, but several others already have. If you are interested in this topic, you can find a lot of information at the links listed. I recommend that you read the first link, in particular. It explains in detail the beliefs of the Separatists (Pilgrims) and the role that faith played in their every day lives.