Yes, the emergence of nation-states, with rulers who decided (even if they remained Catholic) that they could run the church better than the Pope was certainly part of the shift, with Rome responding by becoming ever more authoritarian too.
I think puritanism in a broad sense was a hard-line reaction against the perceived moral breakdown associated with sale of indulgences, paying priests to pray for one's soul, and suchlike insurance policies against the consequences of sin. But Puritanism with a capital P had a particular, Calvinist, theological foundation with belief in predestination, being actually chosen by God not only to be saved (no matter how sinful a wretch one might be), but to impose the Law of Righteousness on the non-chosen sinners (though they were damned to Hell anyway) And, again, it mirrored equally extreme claims to exclusive access to salvation and possession of the moral high ground among some powerful figures in the Catholic church. But there were many other, more moderate, pragmatic, or simply canny shades of opinion.But the new religion could be justified by having a moral highground over the old, so it became more purtianical to form a distinction.