For the past 400 years, the Scientific Method of inquiry, analysis, proof and reproducibility has revolutionized the world. Beginning in the Late Rennaisance, this process brought humankind to deep levels of understanding of how the physical world works, and laid the groundwork for the Industrialization of the World in the following centuries, raising the numbers of individuals and their standard of living, a process that continues today. It also threatened the established philosophies of the day (and the people in institutions that upheld them) as to the role of the divinity in human activity.
Earth and probably the rest of the Universe operates on the principles of cause and effect. In times past, when effects were observed, but not causes, they were attributed to some extrahuman entity, or a cause was invented -- mice come from piles of straw, plagues are God's punishment on the wicked (even when the "good" died right alongside them.)
The fact is that the Scientifc Process answers questions, and shows the relationship between cause and effect. None was more dramatic than Newton quantifying the motion of the physical world ("physics") during the First Scientific Revolution, until Einstien enlarged and enhanced it during the Second Scientific Revolution. Questions that are unanswerable do not prove that science is inadequate or unable to answer the question; it shows that there is a lack of data to support one conclusion or another, that the cause and effect relationship is indeterminate. If there's one thing science is good at, it's proving human conjecture wrong. That we don't have answers to questions today in no way implies we won't have answers tomorrow. In fact, most likely, science in time will answer all questions posed to it; the penalty is that several new questions may be raised, which may or not be currently answerable. But new techniques are developed. New tools are used. New data is available for analysis.
Auguste Comte (1798-1857) once wrote:
To attain a true idea of the nature and composition of this science [astronomy], it is indispensable...to mark the boundaries of the positive knowledge that we are able to gain of the stars....We can never by any means investigate their chemical composition.
That was in 1842. Seventeen years later, Gustav Kirchhoff (1824-1887) relying on the recent work of other scientists, determined the chemical composition of the Sun and other stars through the new science of spectroscopy.
Part of the reason that it is essential to understand that science can and cannot answer specific question is because it allows individuals to see science as one of many tools of human inquiry. There is an all too real tendency to oversimplify thought and rely on one branch to be the totalizing force behind questioning and all discourse. When there is an understanding that science has both possibilities and limitations, a more healthy approach towards the discipline emerges. This can prevent any potential abuses within it and allows individuals to be multi- disciplinary in their approach to solving problems and developing a wider discourse which embraces more people and voices within it.
I am starting from the idea that everyone knows that there are answerable questions that face science, so I'm going to talk about why it's important to be aware that some questions are unanswerable.
First, if we don't know this, we may put too much faith in science. We might think that science will be able to answer all our questions and solve all our problems.
Second, me might wait to take action until science has answered a question, not realizing science cannot. The best example of this would be global warming -- if we do not act until science knows for sure how much warming there will be and what effects it will have, we will never act.
Finally, I think that the disputes between science and religion would calm down a little bit if we realized that science can never answer the question of why the universe exists or whether God exists.