Did the Puritans and the Pilgrims have anything in common?

The Pilgrims were indeed Puritans. This means that they wanted to reform the Church of England radically from within, removing any lingering traces of Roman Catholicism. Put simply, Puritans believed that the English Reformation hadn't gone far enough and that much more needed to be done to ensure that the Church of England achieved the same standards of godliness displayed by Reformed churches in Europe.
The Church and state authorities were profoundly hostile to the Puritans, seeing them as fanatical trouble-makers who represented a potential threat to the political and religious uniformity of the kingdom. As such, they subjected Puritans to a degree of persecution, which though considerably less punitive than that meted out to Roman Catholics, was nonetheless sufficiently intense to force large numbers of Puritans to seek sanctuary across the Atlantic. There, in America, it was hoped that the Puritans would finally be able to establish the kind of godly kingdom they'd always wanted England to be.