First, let's be clear what we are discussing. By "religion", do we mean some specific religion or religions; or religion itself, the idea that there are powers beyond the physical, measurable world? Do we mean the overarching religious "world-view", whereby the non-physical realm is the ultimate explanation for the physical? Or if we are considering specific religions, then which religions?; and are we thinking of canonical scripture, or lay conceptions, or sophisticated theological theory?
As for "science", that could refer to the scientific enterprise, the scientific method, or merely the current (and contingent) body of scientific theory. It might mean the strictly meta-scientific philosophy known as Occam's Razor. Or it might mean some notional scientific "world-view", whereby everything important is presumed to be observable and measurable, and anything which is not is therefore irrelevant or even non-existent.
Surely, given all these possibilities, the bald claim "religion and science are incompatible" is imprecise at best, and simplistic at worst.
If we are talking about the religious world-view and the scientific world-view, then I would have to agree that these are incompatible. These cannot be reconciled, but that is the nature of world-views - they are all-encompassing, and there is no room for more than one. You subscribe to one or the other, or neither.
And then, if we are talking about the literal reality of the Christian Creation story on one hand, and the most recent scientific cosmological theories on the other, then those too are obviously incompatible.
It can also be argued that a belief in an interventionist god would be incompatible the scientific enterprise, as science is predicated on the assumption that the physical universe follows rules, whereas an interventionist god implies that the rules can be broken ad hoc. I have sympathy for this position, but note that the degree to which science would be undermined is proportional to the frequency with which gods intervene.
But if we are talking about the more general religious belief that the universe is the creation of a god or gods, then this position clearly does not conflict with any scientific theory, and can never conflict with any observation that can be made in the physical world. Gods are outside the physical realm and are thus outside the conceivable knowledge of science. If one wishes to dismiss gods on the basis that they are non-scientific, then that dismissal is philosophical, rather than scientific.
In the end, I contend that science and religion are fundamentally "non-overlapping". Science has no purview over the non-physical universe. Religion is sometimes said to have gross effects in the physical world, and when it does, it finds itself in conflict with science. But a minimal religion, in which gods create the universe, and then largely stand aloof, does not have to overlap with the domain of science. This is, indeed, the position towards which religion is largely being forced to retreat by the successes of science. And yet religion will only ever need to retreat so far, and never further, because science has no sway in the non-physical universe.
So... religion and science.... are they compatible? I don't want to be evasive, but the answer depends entirely on what, precisely, is meant by the question. And that is really the best I can do.
