I will use the most common definitions for these two "worlds". The natural world is everything that we can measure or test in some way (whether we are currently able to, or not). The supernatural is everything else (if anything). Straight away, you can see that we cannot measure or test anything supernatural. In other words, we have no way whatsoever to be aware of its existence at all, let alone its effect on anything in the natural world. For this reason, science does not concern itself with the supernatural. It has no reason to claim that there is no such thing as the supernatural, but simply to state that we cannot know anything about it.
In fact, I think it is reasonable to say we will never know anything about the supernatural, the way I have defined it. If we do finally find a way of measuring or testing something "supernatural", then all we have shown is that it was in fact natural after all. In this way, science ignores anything that cannot be measured or tested, but is happy to study it once such a method is discovered.
Putting this together, the way I look at the world is called methodological naturalism. This system deals only with the natural world, but does not deny the possibility of the supernatural world. The system recognizes however that since the supernatural, if it exists, is beyond our capability to detect, we can learn nothing about it. To me, this is the most logical position. Some people go further to state that there in fact isn't anything supernatural, philosophical naturalism. To me this seems to be making an extra claim which is not needed, to achieve the same results.
Natural and supernatural
If there is such a thing as the supernatural, you may wonder if a supernatural agent can cause an effect in the natural world. Indeed, this is usually the premise of an interacting god. With methodological naturalism, there is no need to deny that this is possible. It may be possible, it may not be. However, there is a big problem. Since we cannot test or investigate any supernatural agents, all we have is the event they supposedly caused. And from what we see, the effect looks exactly the same whether a god was involved, a magic crab was involved, a ghost was involved, supernatural particles were involved or indeed nothing supernatural was involved. We only see and can test the natural part.
So say we have an event which appears to be unexplained by our understanding of the natural universe. The most sensible and honest thing to conclude is that we have an unusual, unexplained phenomenon. There is no way of telling whether there is a natural explanation that we do not yet know about, or if was caused by something supernatural. Again, what we actually see would look the same.
Even if we somehow concluded that there was a supernatural explanation, that is as far as it could go. There is no way of knowing which supernatural explanation is correct since we have no way of testing any of them. If someone claims to have a supernatural explanation, unless they can demonstrate how they know this, they are simply speculating. This is often justified by the question, "What else could it be?" This is the classic argument from ignorance fallacy and is not valid.
Speculating, and being satisfied with, a supernatural explanation is not only invalid, it is harmful. It does not increase our understanding at all, and it reduces the incentive of the satisfied person to go out and find other possible natural explanations. It is a trap very easily fallen into, when something strange happens, to leap to your favourite supernatural explanation. But all you are doing is stifling your critical thinking. I'm not saying you are wrong, I am saying you have no reason to think you are right, and could just as well invent a different supernatural explanation and it would be just as unfounded. If you think you have some way of demonstrating something about this supernatural cause, then you should be able to show this to others if you expect them to take it seriously, and by definition it is then natural and not supernatural!