As I popular and important book which I think has not been mentioned on MO as an example of good writing, here is The Nothing That is: A Natural History of Zero Robert Kaplan (Author), Ellen Kaplan (Introduction) (1999). Grothendieck wrote to me in 1982: "The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps ..." (I had mentioned the resistance to the idea of groupoid.) The story is interesting partly because of the resistance to the idea of nothing, which was associated in some minds with the devil, and chaos, and also in the way that a conceptual change had such profound results. One reason for these results was that it led to a notation which better reflected the operations one wanted to do on numbers, particularly multiplication, and the ability to calculate insurance better led, it might be thought, to the prosperity of Venice. All from counting the number of elements in an empty box! But more likely, from the mark of a thumb in sand as a place maker on an abacus, so associated with calculation.
So part of the story of maths is that of conceptual revolutions, which enabled difficult things to become easy, which of course then enables the practice of more difficult things, and that is one of the roles of mathematics.
Many people want to hear of such conceptual revolutions, rather than of the solution of some problem famous for its difficulty, and perhaps also for the difficulty of understanding its significance for the general world.
My own take on the importance of mathematics is that it develops rigorous languages for expression, description, deduction, verification, and calculation. There is also the notion of mathematical structure as a method of modelling the real world.
