Truth, now there's a thing.
Anthony, you have hit the nail on the head with this:
What science does not do is address the possibility that there might be a non-material world.
It can't address this possibility. Non-material world aside, I don't think any of us would go so far as to think science can explain everything, because not everything is measurable, we have gaps in our knowledge, and our own limitations as a species mean that we will not be able to theorise about or measure everything there is in any case. I think where scientists can come across as "arrogant" is in claiming that science holds all the answers, or that science is the only truth. Science is a journey toward the truths of the material realm, a journey of discovery, a seeking for measurable answers; and a good scientist (as noted by many already) will discard a "truth" if it becomes superseded or disproven, and move on to the next theory, looking for ways to test it, seeking to deepen and widen our pool of knowledge and our understanding of the world in which we live. If scientists dismiss as valueless those things they cannot measure, I can understand why people who have a belief in or value the non-material world would get annoyed.
Religion is purportedly a seeking for the truths of the non-material world, including morals and values and the meaning of Liff, and what makes a person a person. What annoys me personally about religious people, though, is when they talk about truth while ignoring facts. Not all religious people do this thank goodness, but there's a large amount of ones who will and do. As noted by others above, there are many who are only interested in the facts that will bolster their own particular beliefs, and other facts can go hang. This is dishonest and self-deceiving, and no seeker of truth should be either of those things if they can help it.
Being human beings, there are those on both sides of this "spectrum" who tend to believe that in discovering a truth, they have discovered the truth. (Don't ask me who said that, someone literary, but its true in both science and religion all too often.)
I don't think there needs to be or can ever be a synthesis of science and religion, but there certainly should be a dialogue between them. Perhaps the resurgence in "mysticism" is part of the pendulum movement that always accompanies change, but I think science and those who accept it should always do what they can to diminish the influence and prevalence of "magical thinking". That will always be with us, its part of human nature it seems, and all of us possess it to some degree - I think we will never banish it entirely, nor should we let it run rampant - but after all, what's life without a little magic?
And now I have blurbled myself into aimless oblivion, I should probably go and get a bit of hunny...