As long as I can remember both science and religion have both been a big part of my life. I have never particularly had trouble reconciling the two. A look at the basics premises on which both are built can be enlightening. In the gospel we believe in all true not only revealed truth, but revealed truth has higher priority since it comes from a better source. In science everything has to be based on quantifiable measurements or consistently repeatable observations. Science doesn't necessarily say anything about religion, it is just that spiritual activities are hard to measure with scientific techniques and instruments.
I have wondered if being a religious person has helped me with my science as well as with life in general. Not taking too serious what the experts say is helpful. Progress in science is made by going beyond what has been done in the past and not just rigorously following what past experts have done. When I have a new problem to solve I don't usually start by looking up how the experts do it. Instead I try to come up with a method that will work. Then I look up the standard method for solving the problem. Most of the time the standard method works better than the methods I come up with. However, understanding different methods to solve problems gives me a broader understanding of what is going on and sometimes the alternative methods have their advantages.
People who are not involved in science tend to look at science as being more definitive than those who are involved in science. In school we are taught that science gives us facts about the world. For those involved in science, science is a method or a way of looking at the world more than a collection of facts. The present body of knowledge represents the most self consistent set of observations and interpretation of these observation that has yet been gathered. I find science so enjoyable because it is not finished and there is the possibility of making new discoveries or gaining new insight into previous observations.
I find knowing a little bit of the history of science very enlightening. Probably the first major step in the development of modern science dates back to the ancient Greeks in the form of Euclid's Geometer. A small number of assumptions that seemed reasonable were assumed to be true without proof. Then from these assumptions geometer was built requiring rigorous proof for each step of the way. This methodology has formed the inspiration for all subsequent math and science. The marvel of this method is that so much can be explained and proven with so few initial assumptions. However, it is important to realize that this whole structure is built on the initial assumptions that are not proven and there is no way to know if the initial set of assumptions is complete. However, the transformation of society that has followed the development of math and science shows the power of building up a structure from a few initial assumptions using rigorous logic steps.
Up through the middle ages people tried to explain why things fall. They believed it was the nature of heavy bodies to want to move toward the earth. Light bodies, like fire, wanted to escape the earth. Not much progress was made. The invention of science was to don't ask the philosophical questions like that since they can't be unambiguous answered. Science made enormous progress by ceasing to ask the philosophical questions and only ask those questions that have quantifiable answers. That is only ask those question that have measurable answers like: how fast does an object fall, do different object fall at different rates, what is the equation that can be used to predict their motion, etc. Since science has been so successful some people make the extension that only the quantifiable and measurable things are real. That feelings of self existence, beauty, love, hope, spiritually, etc. are all illusions and don't really exist. Scientific reasoning is not useful for addressing these questions since they involve ideas that science has chosen not to include. The limits of science are particularly evident when it comes to the value judgments such as "is there a purpose in my life?" Science can't answer that question because the answer is not something that is measurable. Of course scientists are humans and need to discover the purpose of their lives for their own peace of mind, but their scientific training won't be particularly helpful in guiding them to an answer.
Although spiritual experiences can not be measured with scientific instruments it does not mean that experimental methods can not be applied. In the Book of Mormon an ancient American prophets encourages the people he was teaching to try an experiment (Alma 32:27-43). He likens the Word of God to a seed. The experiment is to plant this seed in your heart and watch it grow. I have done that experiment and enjoyed the fruits of it in my life.
To me I have found that science and religion have more in common than they have differences. However, there are some things that seem different. An example is in looking for small inconsistencies. I have been working with a satellite system called JPSS. During ground testing I found a small inconsistency between what the sensor produced and the expected results. Further investigation by the instrument vendors revealed a problem with the electronics that would have produced a larger error on orbit. In science and engineering looking for these small inconsistencies can produce big payoffs. Another example is showing that the earth is spherical from small inconsistencies in the appearance of ships as they approach the shore (the highest point on the ship is visible first).
In religion on the other hand I have found that looking for small inconsistencies is not usually very useful. Occasionally it can lead to increased understanding, but generally it just leads to an "exercise in splitting hairs." The important truths of the gospel are in the big picture while the finer details are usually in the yet to be revealed category. Different Bible writers and modern prophets are talking to people of different times and cultures so it is not surprising that they sometimes come across differently. For example the Old Testament often uses "fear of God" to mean keeping the Lord's commandments, where we now emphasize keeping the commandments out of love for the Lord. Describing the effects of the Holy Ghost to someone who has not felt its influence is also difficult. For example a burning in the bosom and a calm peaceful feeling sounds pretty contradictory but is actually more a description of different aspects of the same thing than a contradiction.