Jonathan Miller, I think, once spoke as an atheist about the power, the richness, of the metaphor of a God who enters His own creation to experience it as one of the created. It is, of course, much more than a metaphor for us, but understanding how powerful a metaphor God-made-man might be to an atheist should help us understand a primary difference between those who believe in the Incarnation and those who don't.
A God who has never been Man is impossibly distant from us. He is everything we believe the omnipotent Deity to be, but infinitely remote. The Creator-created relationship is akin to that of us making an animated plasticene model: the created has no existence except that the will and whim of the Creator has desired it and the created has no means of influencing the Creator. Submission to the will of the Creator becomes proper religion, and acceptance of the Creator's inscrutable will becomes the only response to whatever the world throws at the created.
This may well sound like Islam, but it is also Arianism, the heretical version of Christianity which seems to have so influenced Mohammed. All of Christianity might have ended up like this, were it not for the fact that the gates of Hell will never prevail against God's Church.
The cosmological impact of God becoming part of His creation must have been something like the Big Bang, but it was at Christ's death, not His birth, that the dead were raised and the veil of the Temple was rent. When Christ was born, it was in an inn, and the only people who realised were shepherds and foreigners. There was a chosen people before, but now all of Humanity was let in on the secret.
There are still Arians: Jehovah's Witnesses for example. Arianism also allows other fanciful beliefs to propagate: that of particular human prophets sent as messengers of God: Joseph Smith, for example, who founded Mormonism. (And what about people, clerics even, who believe that they can change the Church's teachings?)
It isn't hard to see Islam in this context: a misunderstanding of the nature of God leading to a catastrophically poor misunderstanding of the relationship between God and His creation. If God was Man, the distance between Creator and created disappears; if God was Man, we can appeal to Him in his Omnipotence as an equal; if God was Man, we can want what He wants, and He can understand how and why our wants have been perverted from what they should be, and He can nudge us back towards his path. If God was Man, we can relate to Him, and He can relate to us; and that means that we can have a dialogue: not a dialogue of equals, because God-made-man is still God, but a dialogue, because God-made-man is man.
That link, and the existence of the Church God-made-man founded when he physically left us, is what makes our religion so different. We can touch God because he gave Himself to us. He will forgive us when we confess our failings because He understands us as individuals. He is in all of us and our reward, if we merit it, will not be simply to have the best of what is human, but to become part of what He is.