What was religion tied to in England that people wanted to get away from and create a new life?

Unlike the United States and some other countries, England had and still has an official state religion. Depending on the period in question, England was either Catholic or Anglican, until it finally stabilized under Anglicanism.  The Anglican Church is the official church of England, and even the prime minister must be Anglican to serve in that position. (In the United States, Anglicans are called Episcopalians, and the Episcopalian Church is not an "official" church because we have none.)

Given that England had an official religion, everyone who was in charge of anything was Anglican. The king or queen was always Anglican, for example, as were most royalty and members of Parliament. You can imagine that people who were not Anglican did not fare very well.  This religious intolerance set the stage for English settlement of the America.

When it came time to establish a government of the United States, the founding fathers of the country realized what a bad idea it was to have an official religion.  Some of the colonies had had official religions, while others had made a point of religious tolerance.  After the United States Constitution was passed by the colonies, the next step was the Bill of Rights, which has an amendment guaranteeing freedom of religion.  In fact, it is the First Amendment, demonstrating how important freedom of religion, among other freedoms, was to the people. 

In countries that have an official state religion, religious intolerance is more likely to occur.  The religious intolerance experienced by our ancestors in England (and in other countries) led to what I believe is a great American principle.