It is a somewhat philosophical question. I personally believe that the importance of math is due to its usefulness and lots of applications. Mathematics is used in everywhere nowadays; as Ian Stewart said in his book Letters to a young mathematicians, if we put 'MATH IN IT' stickers on everything that used math, then it will be hard to find a thing without the sticker. In this point of view, math is important because of its useful results.
However, some of the most sophisticated results of math, such as Fermat's last theorem, is not that useful or applicable in other fields. Despite their seemingly unusefulness, I think these results are also important because they have extended the boundaries of human knowledge. Knowing is important because it gives us some answers about who we are and how the universe is like. In the ancient times, people thought that the earth is flat and is the center of the whole universe. But now we know that the earth is round, and we are probably not the major characters in the history of universe. In this sense, mathematics is important because it gives us truths.
Now let me give you my personal anecdote. I was occasionally talking about the proof assistant Coq to my analysis professor. Of course Coq cannot prove everything, but she asked me whether I want all the problems in math to be solved by computers, supposing the proofs of computers are just like those of human mathematicians that no one can distinguish the proofs by computers from those of humans. I said yes, since I thought whether it is done by computers or humans, knowing the truth is important. But my professor said she don't like it. She thinks proofs made by computers are not as beautiful as those of humans. She thinks the importance of mathematics is in the joy of solving problems, and the beauty of mathematics comes from the efforts put in when solving problems. In this point of view, unknown mathematical truths are something like undisturbed nature that we have to prevent computers from pervading mathematical world.
Now I am confused. Where does the importance of math come from? Is using computers to prove theorems damage the very nature of mathematics?