There is some confusion among Reformed academics - theologians and church historians - as to how to define who was and who wasn't a Puritan; and whether there was any such thing as Puritans at all to begin with (some actually wonder this).
I see it this way: my observation that there is an academic approach to the faith and a spiritual warfare approach plays into this problem they have. The academic types can't see the spiritual warfare types or their approach.
The Puritans did indeed exist - in the past and today as well - and it should be added that Christians from the time of the Apostles who held these distinctives tended to be called names like 'puritan'; for instance, cathari is the Greek version of the puritan epithet, their history crazily rewritten by their murderers, like academics and popular culture today attempt to rewrite the history of the English, European, and American Puritans, if not, as mentioned, denying their existence altogether. (Look at this John Owen quote from the 1600s: “I will pass over other similar monstrous lies with the simple comment that they all pale into insignificance in comparison with the slanders that the Roman pontiffs have dreamed up against the Albigenses [Cathars], the Waldenses, and other faithful servants of Christ.” [pg. 147 of his Biblical Theology]) Here are the Puritan distinctives:
1. Bible oriented. Bible-believing, Bible-focused, Word of God valuing Christians.
2. They understood the fact and reality of supernatural regeneration by the Word and the Spirit (and overall recognized the work of the Holy Spirit Himself, especially in the foundational areas of regeneration and the preservation of the pure and whole word of God down through time, in a way that always seems to anger mainstream establishment Christianity).
3. They understood the difference between fearing the world and fearing God alone; and that when you fear God alone you don't then fear man or man's opinion of you which enables you to move in the direction of wisdom. They feared God alone.
4. They took a spiritual warfare approach to the faith. They understood and experienced the spiritual battlefield ('Faith hath a piercing eye, to see into the spiritual realm.'). For them this made biblical doctrine actual armor of God. They wanted real armor, hence they had no problem with 'hard truth' biblical doctrine (Calvinism), because it re-oriented them internally to being God-centered rather than man-centered or, in other words, being conformed to Christ.
5. They were practical with the faith (or "reduced to practice" the Christian faith). A soldier on a battlefield is a practical individual. Life and death is on the line constantly. For Puritans the Word of God and biblical doctrine is not merely philosophical or theoretical, but as practical as a spade, a weapon, a fox hole, or a good pair of boots. They also knew you have to practice the faith (summed up in the two great commandments of Jesus) in real time (i.e. be awake to the reality and presence of God always in real time; and love your enemy as yourself, because we always target our resentment at human beings one way or another, and it's resentment, rather than gratitude, that keeps us under the power of our fallen nature), in the traffic of one's average day, to increase understanding of the faith; i.e., to be truly conformed to the image of Christ.
6. Puritans were anti-establishment; or just by their nature outside any and all establishments. They were separated out from the world. They tended to be political targets of religious establishments and objects of mockery to the establishment.
7. Without being academic in the usual shallow ways (while still being willing to exploit any and all influences and sources of on-the-mark teaching, and being grateful for the effort to produce it, while producing it themselves as well) Puritans sought a complete understanding of the faith. They sought parts-in-relation-to-the-whole understanding of the Bible and its doctrine. They knew a Christian is to be a prophet, priest, and king (not an eternal infant in a nursery), and that the bar is raised high to be that, yet the Holy Spirit enables the Christian to meet and exceed that bar. For Puritans learning is active, and individual (we face death and our judgment, ultimately, standing solely on our own two feet).
8. They had a strong doctrine of sin and the very real wrath of God. They knew their own state. Tyndale's metaphor of the venomous snake described the Puritan understanding. We are snakes with poison in us, and we can't get the poison out of us. Only God can. And even if we don't strike with our fangs, it is nevertheless our nature to strike. So from birth, due to original and then active sin, we are by our very constitution unable to be in the Kingdom of God. It takes an act of God to change us, remove the venom from us (give us a new heart), and recognize the righteousness of Christ in us which we appropriate by faith in the life and death of Jesus Christ. In other words we can't improve ourselves enough to get into the Kingdom of God. The leopard can't change his spots. Only God can change us. And until He does, by an act of pure grace, we are children of wrath fit for the lake of fire. This stark realization Puritans came to know at an experiential level.
9. Which gets back starkly to the Bible. The word of God. Puritans knew regeneration was solely an act of God, we can't effect it. Yet the word of God, the living, quickening language of the Bible, is the wild card. God says in His Word, several times: move towards Me, and I'll move towards you. The Puritans knew we move towards God not by ritual or physical buildings, but by reading and getting understanding of the Word of God and by prayer.