The original home of the Pilgrims was England. Unfortunately, there was only one legal way to worship in England. This was the Church of England. You could only practice religion in the ways that this church approved. The Pilgrims took their congregation underground because they were labeled (along with other religious groups) as Separatists. In effect they were outlaws in their own country just because of their religious beliefs.
Eventually they would have been caught and imprisoned or put to death with the charge of sedition. Amsterdam was a center of religious tolerance in this time period. People from all over Europe immigrated to Amsterdam for the ability to worship as they pleased. This is what the Pilgrims did. Yes they where allowed to have church services in the style that they wanted, but as time would tell, they still needed a new home.
A great majority of the Pilgrims were farmers when they lived in England. Once they moved to Amsterdam, which was an urban center of trade, they had to work in other industries. The textile industry was big business in Amsterdam at this time. This meant that many of the Pilgrims went to work in textile factories, this even included the children. They didn't have any choice but to work at whatever job they could find. Since they had to sneak out of England, most of them didn't have very many possessions or money saved. As with any large city that has a large number of people constantly arriving from other places, if you won't do the job, they will just find someone "fresh off the boat" that will do it. The pace of working in the factories was staggering to the Pilgrims. They were by no means a lazy group of people, but when you live as a farmer, you are very busy during spring planting and the fall harvest, but you have a lot of down time during the winter. In a factory, this wasn't the case. Most worked at least 12 hours a day, 6 days a week.
The biggest problem with living in Amsterdam was a cultural issue. Even though they didn't want to be part of the Church of England, they still considered themselves English citizens. The longer they stayed in their new home the more they lost their cultural heritage. It was especially evident in their children. They were growing up more Dutch than English. The Pilgrims felt that it was time for them to move to a new home. The New World seemed like the answer to all of their problems.
They hooked up with a group of people that would provide the funding needed for them to make the 3000 mile journey across the Atlantic ocean. They could repay the loan by working in the New World. In this new land they would be able to carve out a settlement and live by the standards that they thought were important. They didn't have to give up their English heritage, yet they would have religious freedom.
It was a difficult journey at every step of the way. They got cheated and taken advantage of several times. There is even some evidence that the Dutch government had something to do with making their departure date in the fall of 1620 instead of early spring. Instead of starting their new lives in a warm climate able to plant crops, they landed in a cold, inhospitable area and barely made it through the first winter. They endured, and survived. They built a colony and were able to worship the way they pleased and raise their children the way they wanted.