To me, science addresses ideas that are falsifiable. Most core religious tenets, such as virgin birth, are not falsifiable. It is a logical fallacy to interpret absence of evidence as evidence of absence, or, put another way, to interpret non-falsifiability as evidence of falsehood. So, while science can clearly predict how pregnancies can occur (e.g., with non-virgins), and can predict that–given ordinary circumstances–pregnancy will not occur in other situations (e.g., with virgins), the entire premise of the miracle and object of faith is that the circumstances were not ordinary. Science will never be able to tell us anything about theology until we find a way to subject God to the scientific method. So, I don’t see how science contradicts religion.
As far as religion contradicting science, the only times I’ve seen that are when a person is interpreting both and saying that the other is wrong. My own faith isn’t in the people of my faith but in the deity, which I might understand better through other people but which cannot be limited by the understanding of us mortals. I think most apparent contradictions arise from misunderstanding of the other side, much like most all interpersonal conflict. It is hard, after all, to study and fully comprehend a single topic, much less two at the same time.
On the other hand, I do see how science and religion complement each other. For example, religion generally cannot help us build a better widget, while science can; but science cannot help us determine whether building a better widget is a good idea, while religion can. If that’s too abstract, take out “widget” and replace “nuclear warhead” or “genome.”
My logical side says that both science and religion are great within their respective scopes, but their ideas are too broad and deep for any individual to fully consider in practical application. Even the most intent practitioner of either will necessarily shortcut their understanding of their actions with either the alternative or its imitation. So it is better to use the time- and experimentally-tested shorthand offered by both religion and science in our daily lives rather than hoping that our off-the-cuff “common sense” will help us muddle through acceptably well.
I think most people who reject science for religious reasons owe it to themselves to learn more about what science is. Likewise, people who reject religion due to the apparent conflict with science owe it to themselves to learn more about religion. There are both religious crackpots and scientific crackpots, and it isn’t rational to lump them in together with their more genuine counterparts, but it is impossible to tell them apart without knowing about the fundamental ideas of either.