First mathematical problems were found in astronomy, commerce, taxation and land measurement. And all mathematical researches help developing other sciences. And reversal, mathematical knowledge that was used in one area proves useful in other areas, even unrelated. And what's most surprising - the purest mathematics often have practical application in engineering and natural sciences.
There is a big difference between pure mathematics and applied mathematics. But as we said, sometimes first one turns out to be applied. Also it's typical for mathematical knowledge to bring to birth other disciplines, like statistics, computer science and operations research. In other words adjuvant part of mathematics become independent sciences, although closele related to mathematics.
In what concerns aesthetics, many scientists often claim that mathematics has certain inner beauty and the most valued qualities are generality and simplicity.