Both. The Puritans wanted to purify the Church of England, reform it. They mostly thought of reform as being in the direction of Jean Calvin and presbyterianism, away from what they saw as Catholic trappings- like bishops, and fancy altar rails and screens behind which the priests did their important transactions with God. And they were opposed by Anglicans like Archbishop of Canterbury William Laud, who liked altar screens, and both kings James I and Charles I, who liked bishops. It was those Puritans who created the Boston Bay Colony. But the Brownists, who created the earlier Plymouth Colony ( now known as The Pilgrims), were separatists. They also had in mind a more Calvinist church, but instead of wanting to reform the Church of England from within they wanted to meet and worship apart from it. Believe it or not, the Brownists were considered more radical than the Puritans. At this time most people thought there was only one church to which everyone should belong. The idea that a group could simply follow their consciences in deciding what to believe was very disturbing, and before fleeing to Holland and then New England the Brownists were often arrested, jailed, and beaten.

The Puritans included some quite important people, aristocrats like Lord Saye and Sele, the Earl of Lincoln, and Sir Harry Vane. The Brownists were not, often were quite poor. And the aspirations and paths of the two groups in moving to the New World were also different. The Brownists had been very much driven out of England to Holland. There they managed to eke out a basic living and were more free from persecution. But they still had some difficult relations with the Dutch Calvinists, the adults had problems learning the Dutch language, and they felt they were losing their English identity- in short, they were an immigrant group that didn't want to assimilate. They therefore were looking for a secure place to settle. London merchant Thomas Weston was looking to revive the long dormant Virginia Company, and needed colonists. The Brownists stepped up.

The Boston Bay Colony was a different venture. It was well-organized, better funded. Its Puritan leaders were well-connected. They had a goal to create something like St. Augustine's City of God, a place where their religious reforms could be put in place. These and the ones who emigrated- John Winthrop, Lady Arabella Fiennes, Charles Fiennes- would also be the leaders in the Colony. Democracy was not a goal: these may have been religious reformers, but they were anything but progressive politically.

Mixed into both of these groups, however, were people who had slight interest in religious issues: they just were looking for a livelihood. England of 1620 had, like all Europe, an agricultural economy and had exploited most all of its land. That left a large, landless workforce with nothing to do. That large workforce, male and female, were recruited for both Plymouth and Boston Bay. They were in both cases very much needed: all the North Atlantic colonies were hardscrabble affairs, and needed all the help they could get. The Brownists called themselves Saints, and the non-Brownist emigrants Strangers, and these are good terms to use for both colonies.

Once the common historical narrative would have focused on the Saints, and say both colonies were motivated by religion. Then more attention was paid to the Strangers, and the simple economic reasons for going to the New World. The Strangers did a lot of the heavy lifting of making these fragile little settlements possible. If no Strangers had signed on, it's possible, even likely, that the two colonies would not have been attempted. Around 20,000 people would emigrate to New England in the Great Migration of the 17th c., and there must have been a lot of Strangers in that number. There's a problem of religion and economics being very hard to separate in this period, so many colonists would have what we'd call mixed motives. It's doubtful this question will ever be settled. But we can pretty easily say that the leaders of both colonies had strong religious motives for going there.

Of course, once they landed in the New World there was little real distinction between a Calvinist group that wanted to reform the Church of England that wasn't there, and a Calvinist group that didn't want to reform the Church of England that wasn't there. The bigger Boston Bay Colony would eventually absorb Plymouth.

Bailyn, B. (2013). The Barbarous Years: The Peopling of British North America--The Conflict of Civilizations, 1600–1675 (1st ed.). Vintage

Bunker, N. (2011). Making Haste from Babylon: The Mayflower Pilgrims and Their World: A New History (Illustrated ed.). Vintage.

Morgan, E. (2006). The Puritan Dilemma: The Story of John Winthrop (Library of American Biography) (3rd ed.). Pearson.

As unsatisfactory as this answer can feel, yes and no.

There's a common misconception that the Puritans got on the Mayflower in England and sailed right to the US, but that isn't true. Essentially, the Puritans that would eventually be called the Pilgrims formed their own congregation that was more or less aligned against the church of England at the time. Their belief was that churches should be voluntary and democratic rather than wholly Christian.

This sounds strange to us today, but at the time, the Act of Uniformity made it so that not attending the official Church of England services was illegal. Missing service accrued a fine and large enough fines led to imprisonment, with one of the larger crimes being conducting unofficial services, which Puritans tended to do.

To them, this was a lack of religious freedom, and so they formed the Separatists. Not a Star Wars reference but the group separating from the church, led by three Puritans named John Robinson, Richard Clyfton, and John Smyth. There was another man named William Brewster, who we'll get into later. The Separatists formed congregations in Nottinghamshire and hoped that the rule of James I in the early 1600's would bring change. Instead, it didn't, and their request for supplies was only granted for an updated Bible. The Archbishop of Canterbury began prosecuting them, and before long, most Separatists were confronted, fined, and imprisoned. Most consider that Archbishop's successor, Tobias Matthew, the one to credit for removing non-Christians from England.

"But after these things they could not long continue in any peaceable condition, but were hunted & persecuted on every side... For some they were taken & clapt up in prison, others had their houses besett & watch night and day, & hardly escaped their hands; and the most were faine to file & leave their howses & habitation, and their means of livelehood." - William Bradford, "Of Plymouth Plantation."

Translation, the Puritans/Separatists/Pilgrims (all the same thing) bailed because the only other thing they could do was stay and get arrested for not being Christian. So did they leave England for religious freedom? Yes. Did they go to the Americas for religious freedom? Nah, not really.

Instead, the Pilgrims went to the Netherlands - Leiden, specifically - in about 1608. One Puritan leader, William Brewster, had been the presiding elder for the congregation, accruing the modern equivalent of thousands of dollars in debt for not complying with England's laws. In Holland, most of these Puritans were able to find work in common trades such as textiles, printing, brewing, and education. Others struggled through the language barrier or a lack of marketable skills. One member of the congregation purchased a sizeable estate for these Puritans to live on, as well as holding religious/political debates, which William Brewster began to print and circulate with typesetting equipment via a local press.

Thing is, the Dutch had an incredibly different culture from the English Puritans, and this is where the belief that they were religious extremists come in. The Dutch were much "looser" morally than the Puritans, which the above-quoted William Bradford had thus to say about:

"But that which was more lamentable, and of all sorrows most heavy to be borne, was that many of their children, by these occasions, and the great licentiousness of youth in that country and the manifold temptations of the place, were drawn away by evil examples into extravagant and dangerous courses" -Bradford, Of Plymouth Plantation.

Translation again - our chaste, goodly Puritan citizens are being tempted into the Dutch way of life, and we don't like that. There were other complications - Holland was in revolution against Spain at the time in some parts of the nation, and were extremely close to revolution in others. The nation was in turmoil economically and politically, and the Puritans feared that this wasn't a place to raise their children, especially given the mismatched cultural values they shared with Holland.

Some saw the Americas as a place of Providence as well - Providence essentially being their destiny and God's blessing - which placed extra pressure on Brewster to lead them there. To make matters worse, Brewster also had his printing-press hands in religious unrest in Scotland - then still firmly linked with England. This led to him being wanted by authorities, his printing press seized, his financer arrested for a 14-year prison term, and himself evading authorities repeatedly.

There were other places that they considered going to instead of the Americas. The Dutch had a colony in South American Guiana, for example, that they considered. Ultimately, the thought of settling near one of the Virginian settlements (yes, there were already settlements before the Pilgrims) won out. They left in several different ships on several different occasions rather than all at once for financing, room, and supplies reasons. The largest ship, the Mayflower, was a lease. Sometime later they arrive in Cape Cod, sail around the cape towards the Hudson, turn around because that part sucks, and anchor the ship near Provincetown Harbor in November of 1620. This is all extra information to show that the Americas weren't their destined land but a choice they made off of budgeting and political pressures.

So, in summary, the Puritans left England for religious freedom and went to Holland. There, by comparison, they were more or less religious extremists, and left to preserve their religion and way of life, as well as to avoid political/religious/militaristic unrest. In essence, they were both - refugees that wanted religious freedom and extremists that wanted all these dang Dutch prostitutes out of their way and gone from their children's sight.

If you're interested to know more, I strongly support reading William Bradford's "Of Plymouth Plantation," which I cited twice here. It's the story of the Pilgrims from 1608 up until about 1647/48, written by the leader of the Plymouth Colony, William Bradford (duh). I have a Masters in English Literature, and I can't begin to explain to you how often I've referred to this journal academically.

Hope that answers your question, and I hope this post is acceptable, mods! This is my first post here - I got giddy when someone asked something a literature degree could answer.