Absolutely nothing in /u/Gadarn's comment is incorrect.

"Puritan" is initially a pejorative term for people critical of the post-Elizabethan religious settlement of the Anglican church. It has insulting overtones; early alternate terms are "precisian" or "dissenter," which connote a kind of stick in the mud self-righteous piousness that insists on minor doctrinal points.

By the start of the English Civil War the term entirely falls out of use in favor of newer insults specific to politics (e.g. "roundhead") or a preference for more precise description of theological adherence (e.g. Presbyterian, Independent, Anabaptist.) There's a tremendous amount of debate about how useful the word puritan even is to describing the religious politics of the early 17th c., some historians have argued we shouldn't be using it at all because its meaning shifted so much and it was never really a self-identifier.

So for the rest of my answer we'll accept the frame of your question, that there's an identifiable population called 'puritan,' but what I want to emphasize is that people we call puritans aren't remarkable for anything other than their religious beliefs. One doesn't get called a puritan by their neighbors for dressing modestly-- many pious people do so-- but what's puritanical is a Calvinist belief that causes one to stop attending Anglican services and worship differently.

What I've hope we've established is that the puritans consider themselves simply to be properly Reformed Christians. They would refer to themselves in those terms, as reformed, "godly," as the Calvinist "elect," or as modern "saints" of the true Christian Church. Everything that distinguishes puritans in terms of their behavior comes out of these beliefs.

Take sex, which prompted your question, as an example. It's true that Calvinists/puritans see marriage more as a divinely instituted arrangement than a socially/legally mediated one; they have a marriage theology that prioritizes a couple's sexual union as understood Biblically over the authority of the Church of England. The source of their controversy with society's preexisting marriage norms is deeply rooted in their Calvinist attitude that elected Christians can and must understand their religion through Holy Spirit guided Bible study, and that the understanding they arrive at is superior to secular authority.

I'm going to attach below an answer on English puritanism I've provided here before. The purpose is to explain Calvinism's appeal and who it appealed to.

BREAK

After the defeat of Protestantism in France's religious wars England receives large numbers of Huguenot refugees. Tens of thousands of French Calvinists streaming in to England's cities and large towns in modern Britain's first influx of peaceful immigration. The refugees introduce Calvinist theology to England and there's an explosion of dissenting/reforming Protestantism over the next decades. Who did Calvinism appeal to in England?

Calvinism appealed most to the 'middle sort of people' or 'industrious people,' reflected the concerns they had about society, and provided a vehicle for self-affirmation/class organization.

I want to break that down. By middle sort/industrious I'm talking about a group distinct from the modern concept of a middle class. These are gentry (non-noble landowners, usually with tenants); established artisans (who have workshops employing journeymen); tradespeople; and certain civil functionaries like lawyers/clergy. Lump in people who aspire to these positions as well, who imitate the Calvinism of their idols.

The social concerns that Calvinism addressed were these:

1. Continued widespread corruption in the Church. The English Reformation took the monasteries and their land away from the Catholic Church but did not, as Protestant ideologues like Luther wanted, give it to the poor. Instead the land was appropriated to enrich the nobility and their hangers on. Ecclesiastic courts are another major irritant for this group. They are accused, with some justice, of bringing cases for offenses like adultery primarily against these middle people. The fees, fines, and legal costs that these middling sort bring in create an incentive structure for the courts to prosecute them; the poor by definition cannot pay anything and the aristocracy is superior in practical authority.

2. A monarchy that seemed unaccountable and amoral. This is complex, but broadly speaking many Protestants found the Stuart monarchy to be actively impeding the Christian mission, for instance by allowing most entertainments on Sundays. James I was known for conspicuous consumption on a grand scale. Under the direction of literary great Ben Jonson James I staged masques that could cost thousands of pounds (a very great lord's yearly income.) Charles I toned down court celebrations considerably but still managed to collect some of Europe's finest art.

3. A highly hierarchical Church in which only the least Protestant clergy could advance. This ties into 2; the religious appointments and policies of the crown (Arminian, sacramentalist) were viewed as being too close to Catholicism and chosen to legitimize the sinfulness of society. These policies were promulgated and enforced by Archbishop Laud, who would soon meet a bad end at Puritan hands during the Civil War.

4. The 'middling sort' were a group who had attained success but were still in danger of penury. The doctrine of election corresponded to their precarious positions and inspired a self-confidence based on moral superiority to a dissolute aristocracy or hedonistic lower class.

5. Readers of the English language Bible are drawn to Protestantism. The King James Version increases English readership of the Bible significantly. Personal access to scriptural texts is in itself a revolution in approach that encourages further revolution. Bible readers become critical of official Church teachings and perform their own radical exegesis. Often this morphs into a kind of extreme biblioidolatry-- a conviction that the official church is entirely untrustworthy and only personal/Spirit-assisted reading of the Bible can bring a person to true religion. This can lead to idiosyncratic readings by some individuals, but also new practices like Sabbath-keeping which was a major Puritan mania.

Class organization: Calvinist churches were organized on the principle of election. Councils of lay elders would be responsible for choosing preachers and maintaining church 'discipline.' These elders would usually be economically substantial community leaders. The group of morally and socially elite 'elect' leading a broader 'unregenerate mass' of people whose salvation was uncertain. Contrast this with the official Anglican church, where the priests are appointed by the monarchy and everyone below is lumped together indiscriminately. The middle sort don't want to be considered to be the same as irreligious peasants.

Society and Puritanism in pre-Revolutionary England remains one of the better places to start on 16th century English Protestantism and the arguments I've reproduced now are highly indebted to it.