When discussing the role of religion in the rise of science, it is important to be clear of the time period, the religion, and the science being discussed.
For example, the Western church in late antiquity was instrumental in the decline of 'pagan' learning. The role of astronomers and mathematicians was judged as anti-Christian, mostly because of the association with astrology. But that also meant that the advances in math, mechanics, astronomy, etc were cast aside and denigrated for centuries.
On the other hand, the later church was instrumental in the translation movement (say, 1000-1200 AD) that re-introduced those same pagan ideas back into Christianity. But in so doing, it explicitly excluded certain ideas from consideration (like the eternity of the universe). Many of those who formulated crucial scientific ideas (such as inertia, uniformly accelerated motion, etc) were priests or monks. Of course, that is partly because only the priests and monks were literate at the level to read and discuss the relevant ideas. The idea that Ptolemy, for example, needed to be updated, was widely acknowledged.
We can also contrast medieval Christianity with medieval Islam. Up until about 1200-1300 AD, the Islamic world was the center of intellectual advancement. Al-Haytham produced the most important study of optics prior to Newton. Mathematics flourished. So why did the scientific revolution NOT happen in Baghdad? At least part of the reason (and there were many) was the idea that became current that any such investigation was 'alien' and un-Islamic. The investigations that did happen occurred outside of the official school system. Also, there was a concept that God could do *anything* at any time and so any patterns that might be found were unreliable. Hence, the very basis of science was denied.
In contrast, by the 14th century in Europe, there was a feeling both that 'ancient learning' needed to be reconciled with religious faith, but also that it was *possible* for mere humans to learn about how the universe works. This idea, while classical in spirit, became a religious default. Add to this the intellectual ferment of the schools to decide which ancient teachings were valid and which were invalid, and there was a clear *religious* incentive to explore scientific ideas.
Then the Reformation came along. The split in Christianity had a couple of consequences. One was that the Catholic church became much less tolerant of new ideas. Bruno and Galileo were victims, in part, of this hardening of dogmatic lines. But it also made it so that in *other* areas, under the Protestants, early scientific investigations could be done with some impunity. So, once Galileo's ideas made their way out of Italy (which happened quickly), they were tested and extended by those in Protestant Europe. Many of the scientists from this period were strongly influenced by their faiths (including Newton, for example).
This identification of Protestants and science lasted until about the time when investigations into geology showed that the Biblical stories were simply not consistent with the facts on the ground. At this point, and even more when Darwin's ideas became well-known, the religious backlash was severe in certain camps. To some extent, this applies today; the creationists ultimately look back to this time period for most of their ideas.
On the other hand, by this point the Catholic church had realized a good deal of its errors and had started doing more science again. So, the development of cosmology and the Big Bang scenario was done partly by a catholic priest (Lemaitre). The Vatican astronomers are still some of the best in the world.
So, it is easy to point to various times in history when religion and science *were* at odds, there are also many times when they walked hand-in-hand. The point is that religion *can* be a motivator to explore the science, but it can *also* be a significant hindrance when that science leads to ideas the religion finds intolerable.
