To begin, who were the Puritans? The Puritans were a movement within the Church of England to further reform the church along Genevan lines. They had three rough concerns (according to J.I. Packer): a) concerns about prayer book ceremonies; b) concerns about the lack of good preaching; c) concerns about forms of church government. All Puritans were concerned about a) and b) but not all were so about c). Within the last category there were episcopal Puritans, i.e. those who were content with the government of the Church as it was; there were presbyterian Puritans, i.e. those who wanted a government similar to that of Geneva; and there were congregationalist Puritans, i.e. those who wanted the Church to be governed only in the congregational level. All of these were part of the Church of England at one point and eventually, the presbyterians and congregationalists drifted away, but episcopal Puritans stayed behind in the Established Church to further "purify" it, hence the name. J.I. Packer says,
‘Puritan’ as a name was, in fact, mud from the start. Coined in the early 1560′s, it was always a satirical smear word implying peevishness, censoriousness, conceit, and a measure of hypocrisy, over and above its basic implication of religiously motivated discontent with what was seen as Elizabeth’s Laodicean and compromising Church of England. Later, the word gained the further, political connotation of being against the Stuart monarchy and for some sort of republicanism; its primary reference, however, was still to what was seen as an odd, furious, and ugly form of Protestant religion.Puritans in the 16th century were also known as that "hotter sort of Protestant."
The Puritans were concerned about a few of the ceremonies allowed in the Prayer Book which they regarded as "popish". These included: a) the giving of rings in weddings; b) kneeling to receive Holy Communion; c) the wearing of the surplice; d) the sign of the cross in baptism. There were other scruples but these were the essential ones. Briefly, a) implied a sacramental theology of marriage to the Puritans (even though Article 25 expressly denies this), b) implied an adoration of the elements in the Sacrament (again expressly denied in the Black Rubric), c) implied that English priests were sacerdotal priests, standing between God and man, and d) was regarded as superfluous to the action of baptizing in water. In response to liturgical forms of worship, there was also a division among the Puritans, "There were, and are, three possible ways of ordering public worship: to have a set liturgy like the Book of Common Prayer, or a manual of general guidance like the Westminster Directory, or to leave it entirely to the individual minister or congregation to regulate its own worship at will. "
All this to establish briefly who the Puritans were and what their objectives were. It is interesting to note that the Puritans did not have many doctrinal scruples with the English Church. J.I. Packer notes that William Laud was just as Reformed as the Puritans in theology but in matters of ceremonial and worship, the parties diverged. Interestingly enough, church polity was a matter of doctrine for the Puritan party but not for Anglicans at large, who (at that time), regarded church polity as a matter of adiaphora, thus holding to the bene esse position of the episcopacy in the life of the Church.
With this "Anglican Myths" I wanted to briefly introduce the Puritans and remind us that they were Anglicans. There was no artificial separation as there is now in talking of "Anglicans and Puritans". Puritans were members of the Church of England who wanted to further "reform" it along Genevan lines (obviously a point I wouldn't agree with but nonetheless).