This question has many answers in many different contexts. Two of my favorite writers, G.K. Chesterton and Soren Kierkegaard, focus on the difference Christ makes in terms of human possibility.
Man is different from other animals insofar as he lives self-reflected in a world. Beavers and dogs don't worry about how they relate to the world; they just exist as they are unselfconsciously in the world. They are the world. But man knows himself as who he is in relation to the world. Kierkegaard describes this difference in The Sickness Unto Death in terms of the self as "a relation which relates itself to itself." The fact that man by nature relates himself to the world means his existence, unlike that of non-rational animals, is a dialectic of possibility and necessity. I understand who I am (or think I understand), and I also understand the world and my place in it, and in terms of that relationship life presents a present reality of necessity and a horizon of possibility. I exist as a relationship to the world, but I can know that relationship and (perhaps) change it - I can relate myself to the relationship which constitutes my self in the world.
But I can do that only in terms of the possibilities available to me, and those are constituted by my philosophy. What sort of possibilities are available to the natural but pre-Christian man, that is, the pagan man? Chesterton in Orthodoxy describes the pagan world as a world of pink. The great pagan virtue is moderation; a little of everything but not too much of anything. Red and white mixed together, not too much of each. This is a natural and sensible policy, and in the pagan world it produced great men like Aristotle and Marcus Aurelius. The ideal gentleman is a little bit of a warrior and a bit of a scholar as well. He drinks wine but not too much; he loves others but not too much of that either. For love is a form of madness and madness is unbalanced. Above all he maintains self-control, for he knows that the world contains good things as well as evil things, and that it ends in death. He keeps these facts before him and holds himself well so that he is neither carried away by good fortune, nor destroyed by misfortune, for life inevitably involves both. There is no better wisdom in a world without Christ, especially in a world that cannot imagine Christ. The life of balanced moderation is the best life that the best pagan mind could imagine; it defines the horizon of pagan possibility.
What has changed with Christ? The Gospel of John tells us that His first miracle occurred at Cana, and involved the replenishment of wine at a wedding feast that had run dry. We can assume that the host of the feast had on hand an appropriate amount of wine for the celebrations. It would seem, then, that any additional wine would violate the principle of moderation; we've gone from having a sensible good time to getting drunk in excess. But this is why it is a miracle, for a miracle is more than merely the suspension of ordinary physical expectations; it is a sign and revelation of a new order of existence, an order that breaks through the old pagan compromises and proposes a way of life that answers to the transcendent meaning of Christ. The exhaustion of the wine at Cana symbolizes the exhaustion of pagan virtue and the existential hopes it offered. The party is over; it is expected to be over and the celebrants are prepared to go home; no one can imagine the party continuing, or at least continuing with any propriety. But Christ can imagine it, and through His grace he turns water into wine, that the party may continue, theoretically indefinitely. From that moment forward the horizon of pagan hope has been forever shattered, for the possibility that it is not the final limit, that there is a way of life that is not bound by pagan compromises, has been permanently introduced into the human imagination.
Chesterton describes the difference as a world of pink becoming a world of bold reds and whites; reds for the warriors and whites for the monks. There were warriors in the ancient world, of course, and pacifists as well. But the pure warrior, like the pure pacifist, could not express an ideal human type because he violated the principle of moderation or balance. More significantly, the warrior and the pacifist had nothing to do with each other. Each might despise the other and, if they didn't, by the nature of things they at least expressed different philosophies of life. But in Christendom the martial Knight was as much an expression of the authentic Christian life as was the peaceful Monk. Far from expressing opposite philosophies of life, they both expressed different ways of performing the same mission: Redeeming the world in the name of Christ. Chesterton states the difference this way: In the ancient world the balance of existential possibilities was expressed in the single individual of the moderate, virtuous gentleman. In Christendom, the balance of possibilities occurred in the Church as a whole rather than individuals:
This was the big fact about Christian ethics; the discovery of the new balance. Paganism had been like a pillar of marble, upright because proportioned with symmetry. Christianity was like a huge and ragged and romantic rock, which, though it sways on its pedestal at a a touch, yet, because its exaggerated excrescencies exactly balance each other, is enthroned there for a thousand years. In a Gothic cathedral the columns were all different, but they were all necessary. Every support seemed an accidental and fantastic support; every buttress was a flying buttress. So in Christendom apparent accidents balanced. Becket wore a hair shirt under his gold an crimson, and there is much to be said for the combination; for Becket got the benefit of the hair shirt while the people in the street got the benefit of the crimson and gold. It is at least better than the manner of the modern millionnaire, who has the black and the drab outwardly for others, and the gold next his heart. But the balance was not walkways in one man's body ad in Becket's; the balance was often distributed over the whole body of Christendom. Because a man prayed and fasted on the Northern snows, flowers could be flung at his festival in the Southern cities; and because fanatics drank water on the sands of Syria, men could still drink cider in the orchards of England. This is what makes Christendom at once so much more perplexing and so much more interesting than the Pagan empire; just as Amiens Cathedral is not better but more interesting than the Parthenon. - Orthodoxy, Ch. 6For both Chesterton and SK, the advent of Christ permanently changed the nature of existence and of the world - and that whether you believe in Christ or not. The key point they share in this regard is that Christ revealed possibilities that were unimagined prior to the Incarnation. After the Incarnation, those possibilities cannot be eradicated from the human spirit, even if Christ Himself is later denied. The price of denying Christ cannot be a simple return to the pre-Christian world, for the possibilities he revealed will remain in the human imagination- it is only their fulfillment that will become impossible, since that fulfillment is only possible with the grace of God. The result is that post-Christian life can never be a simple return to paganism; it will instead be one of melancholy and despair.