There are scientists who believe in the Steady State model of cosmology, or that HIV doesn’t cause AIDS, or that sunspots are the primary agent of climate change.
The mere fact that such positions are held by some scientists doesn’t make them good scientific positions.
But that just means they are different, not that they are incompatible.
(Here I am deviating somewhat from Coyne’s take, as I understand it.) An airplane is different from a car, and indeed if you want to get from Los Angeles to San Francisco you would take either an airplane or a car, not both at once.
We shouldn’t declare them to be incompatible purely on the basis of what they are, which some people are tempted to do.
Certainly, science works on the basis of reason and evidence, while religion often appeals to faith (although reason and evidence are by no means absent).
We know more about the natural world now than we did two millennia ago, and we know enough to say that people don’t come back from the dead.
In response, one strategy to assert the compatibility between science and religion has been to take a carving knife to the conventional understanding of “religion,” attempting to remove from its purview all of its claims about the natural world.
We can easily conceive of a world in which the best scientific techniques of evidence-gathering and hypothesis-testing left us with an understanding of the workings of Nature which included the existence of God and/or other supernatural phenomena. Thomas Aquinas, were he alive today, would undoubtedly agree, as would many religious people who actually are alive.) It’s just not the world we live in.
(That’s where they would disagree.) The incompatibility between science and religion also doesn’t mean that a person can’t be religious and be a good scientist.