First, what is "religion"? To frame the polarity as between "science" on the one side and "religion" on the other is to bias the discussion from the start. That's because those who prefer "science" to "religion" are almost the only people who lump all religions into one category called "religion." Few actual religious people do this.
For example, I read part of a book by one partisan of science who lumped together Christian fundamentalism and Islamic fundamentalism. Notice the word order here: to this author, the ideological Other is a thing called "fundamentalism." It happens to have a Christian species and an Islamic species, but fundamentalism is the genus. But very few Christians and Muslims think of it this way. To them there is no Christian fundamentalism and Islamic fundamentalism. Rather there is fundamentalist Christianity and fundamentalist (Wahhabist) Islam. They are two different religions, two different genera. They each have a species that is similar to its counterpart in some way, but those are really quite different animals, like a blue jay and a blue whale. In fact, fundamentalist Christians and fundamentalist Muslims are the least likely in their respective religions to see themselves as proponents of "religion" in general or to see each other as fellow "fundamentalists."
It does no good to talk about the relationship between "science" and "religion" because different religions have different postures toward science. Different tribes and traditions within a particular religion have different postures toward it. For that matter, even the same tribe or tradition may have different postures toward science in different places or at different eras of its existence.
Let's take as an example the popular whipping boy, evangelical Christianity. Evangelical Christianity earned its antiscience reputation in its fundamentalist phase in the early 20th century when Darwinian evolution was the flashpoint. But few people know that in America before the Civil War, evangelical Christianity wasn't just pro-science, it drove science; antebellum developers of "natural history," as it was known then, were motivated by religious belief. Today the posture of evangelical Christianity toward science is muddled and mixed. There is still much fear, suspicion, and disinterest. There are also many evangelicals quietly working in the sciences along with a few luminaries like Francis Collins, who headed up the Human Genome Project and currently directs the National Institutes of Health, and Stamatis Vokos, who is a pioneer in physics education.
The point is that it only confuses matters to talk about the relationship between science and "religion" in general. To do any good you must get much more specific as to what religion you're talking about.
The second clarification needed is, do we mean science or Science?
Lowercase-"s" science is composed of two things: a method and a body of knowledge acquired through that method. Uppercase-"s" Science is also two things: a philosophy and a culture that reflects, reinforces, and passes on that philosophy, which combine to orient their adherents to ultimate things. In other words, Science is a religion. Admittedly, it's a rather unusual religion; it's hard to think of many religions that deny the existence of the supernatural, either ontologically or merely functionally, as a major tenet, or who make rejection of faith an article of faith. (They call it "reason," because most of them have not studied epistemology.) However, it does have points of contact with certain strains of Buddhism, and perhaps Science is another tribe or tradition of the secular religion I wrote about previously. Uppercase-"s" Science's appeal is not as broad today as it was during Isaac Asimov's and Carl Sagan's careers, but it is still very much alive.
I don't know enough about Science to sketch its contours accurately, but I can give you an example of what I mean. There is a marvelous series of electronica songs and videos called Symphony of Science that use "auto-tuned" statements by popular scientific thinkers. My favorite, "We Are Star Dust," features supercool astronomer Neil DeGrasse Tyson "singing," "We are part of this universe; we are in this universe . . . the universe itself exists within us. . . . We are part of the heavens. . . . We are star dust in the highest, exalted way, called by the universe, reaching out to the universe with these methods and tools of science."
Tyson (and the others in the song) are talking about how all heavier elements were "cooked" and built inside stars that later exploded and sent their "enriched guts" flying through the galaxy. These elements gravitated toward each other and collected into planets and the things on those planets, including people. So all the atoms in our bodies were once part of a star—we are star dust.
But note how Tyson expresses this scientific conclusion. "We are part of the heavens," a carefully chosen, old-fashioned term from an age when the sky was more than a physical location but also a spiritual one in people's outlook. "We are star dust in the highest, exalted way." Very complex star dust, certainly; living star dust, indeed. But highest and exalted are statements of ontological value, honor, and significance, categories that do not exist in science. "Called by the universe"—really? The universe as a sentient, deliberate, communicative entity summons humans to "reach out" to it and establish a communicative link—this universe that "exists within us"?
I don't know how far to press these out-of-context comments by Tyson; I don't know how literal and accurate he means them to be. Perhaps he just has a natural gift for poetry that can't help but find an outlet. Maybe he is so driven to interest people in the science that he loves that he uses provocative, figurative speech to seize their attention. Or maybe there is something primal within him that longs for contact with a transcendent being that he doesn't believe in that shapes his language. In any case, the statement about where our atoms came from is science. But how Tyson states it is Science.
One listener commented, "Now, this is art. All songs make you think about future and universe. Science's 'Gospel'." It is Science's praise-and-worship music—actually, I listen to it to worship too.
Naturally, you arrive at quite a different answer if you ask about science and religion versus Science and religion. Answering the question about science and religion means observing religious individuals' comfort employing the scientific method and interest in its results. But answering the question about Science and religion is really about Science and other religions—it's a question of comparative religion.
Now, I expect that many adherents of Science believe that you aren't really committed to science if you don't believe in Science and probably even reject the distinction between science and Science. I think they're wrong on both counts.
In sum, as with every question, answering it starts with defining terms. "What's the relationship between science and religion?" That depends: what do you mean by "religion"? What do you mean by "science"?