Bingley, as W'ON noted, the Pilgrims were 'Puritans' - i.e., a group originally within the C. of E. which favored a more Presbyterian style of Church government, worship style, etc.
From the time of the C. of E. being wrenched away from Rome by Henry VIII, there were constant struggles among various parties as to how far reformation should go. When Henry died, Thomas Cranmer, his archbishop of Canterbury, set about really reforming (very little was allowed by Henry). His first Prayer Book of King Edward VI of 1548, one of the great masterpieces of English, was fairly conservative. Before long the influence of continental reformers, Luther, Calvin, Zwingli etc. spread to England and a number of parties demanded further reform. To try to keep the peace, Cranmer produced in 1552 the second Prayer Book of Edward VI, which reduced the 'popish' features. Even this did not satisfy the most radical reformers, who were becoming known as the Puritans, who wanted a Presbyterian form of church.
The debate was abruptly and nastily ended by the untimely death of Edward VI and the accession of his sister Mary, known to history as Bloody Mary. She had remained Catholic and immediately restored the Roman religion and practice and banned Protestant worship on pain of burning at the stake. She was succeeded after, I think, 8 years by her sister Elizabeth.
Elizabeth I was not, or did not think herself to be, entirely secure on the throne, so she trod warily in matters of religion, doing no more than to switch the country back to Protestantism and the 2nd Prayer Book, but the Puritans continued making a nuisance of themselves, so finally a new edition of the P.B. was issued, making only minor changes. The Puritans kept on pressing for radical reform.
In 1600, Richard Hooker, a C of E priest, published The Laws of Ecclesiastical Polity, another great landmark of English literature as well as a great legal work. This was a defense of the traditional polity and organization of the C of E against the Puritans who had never stopped agitating. It was about this time that many of them decided that they couldn't live under this 'popish' church which refused to cave in to their demands, so they left for Holland, later for America. More fools they. Had they stayed in England and lived long enough, they would have seen the final triumph of the Presbyterian party, when they succeeded in dismantling the C of E under Cromwell.
Anyway, their 'persecution' consisted in being unable to persuade any king or queen or synod or anyone else to re-make the C of E into a Presbyterian system, or to allow nonconformist congregations to enjoy the same status as the established church. Needless to say, when they got to America and were in charge of things, they proceeded to persecute all dissenters from their system. Baptists and Quakers met with much worse fates in New England than Presbyterians had ever suffered in Old England.
