Let's reframe this whole science vs. religion thing.
First let's be explicit about what we mean by "science" in this context. Science isn't some monolithic body of dogma that proclaims "this is the way the world is, whether you like it or not." A scientific attitude says, "This is my best explanation for this phenomenon, but if you can poke a hole in it or show me a better one, I'm listening." That attitude has remarkably expanded what we know about the universe, with obvious technological benefits. This has resulted in a widely accepted set of explanations for things--why objects fall to earth, what causes Parkinson's, how the sun works, etc--that people can generally base their worldviews on without spending their lives recreating those explanations.
So, when it comes to "science vs. religion," what we mean by "science" is "a viewpoint informed by the scientific method and currently accepted scientific explanations for things."
That might seem like a long digression and a really wordy statement of the obvious. But the point is that "science" as described above does not claim that "God does not exist." That's not the kind of claim that science makes. The kind of claim that science does make is this: "According to what we currently understand about how things work, we don't need to postulate the existence of God to explain anything."
In fact, given what we know about how human minds work, we think we can explain why humans might want to postulate the existence of God. The human brain is, at its most basic, a pattern-finding machine. Its job is to take in information from nature, extract patterns from that, and make us better at staying alive by finding meaning in those patterns. This works at many levels, from very basic perception (distinguishing circles from lines) to solving abstract logic puzzles. It's the basis of science: looking for patterns in data and asking why they occur. It's the basis of our intelligence, and based on how we're doing as a species, I'd say we're very good at it.
The problem is that we're so good at finding patterns that we're not very good at not finding them. Even in an entirely random stream of data, we're capable of noticing patterns that crop up by chance. It's what happens when you look at a cloud and think it's almost in the shape of a camel. (It's also, I think, the basis of our appreciation of art, but that's a huge digression here.) Once we're found a pattern, we're wired to understand why that pattern exists, to take advantage of it. But, problem: if we've noticed a spurious pattern, there won't actually be an explanation for it. Nevertheless, our minds create one. It's the basis of superstition. Given enough of these patterns that we can't find repeatable explanations for, we're likely to determine that there must be mysterious forces out there that we don't understand. So, our basic cognitive processes give us a good reason to want to believe in something like a god, whether there is one or not.
All in all, then, science says: we don't need God to answer our questions, but we can explain why belief in God might exist. Neither of those those statements makes the claim "There is no God."
Strictly speaking, this is an agnostic viewpoint. The question of God's existence is neither relevant to its functioning nor something that it tries to answer. It's a tangential issue. While holding this viewpoint, I can be a theist and believe in God, or be an atheist and not believe in God. Contrary to Mort, I can also hold no opinion -- maybe I don't care, have never thought about it, or simply don't know where I stand. But that's no more reasonable a stance than being an agnostic atheist: an a.a. merely says "You can't prove it either way, but given that there's no reason to believe, I don't." Which was, I think Mort's original point.