The first reason why there are both sharps and flats is historical. Equal tempering produces the same pitch for, say, D# and Eb. The same was, however, not true for some of the historical tunings prevalent at the time the very notation became standard.

For more information about this, you can read "How Equal Temperament Destroyed Harmony (and Why you should care)". It explains, for example, that Bach's Wohltemperierte Klavier did not, in fact, refer to the equal tempering, but to a different tempering (whose details I can't exactly recall), but I seem to remember something about slightly less sharp thirds. In fact to summarize that book in a sentence -- he doesn't like overly sharp thirds... and sevenths.

There was a veritable zoo of different tunings, and each involved a different set of compromises. So one produced lovely chords as long as you stayed in root position, whereas another produced slightly less faithful thirds but instead allowed you to play secondary dominants of the relative minor, and so on.

Anyway, the point is that there was a time when playing a third up, and playing a sixth down were not the same thing (modulo an octave), and that's why they used different symbols. But we now use equal temperament, which forces that to be true, to hell with those overly sharp thirds (as compared to natural overtone series).

The second reason has to do with the fact that notes are not really useful on their own. They're useful when considered in relation to other notes. In other words, we're really interested in intervals, rather than individual notes. And when considering intervals, it's customary to begin at a certain note, say, C, and count our way towards some destination note. Once you start doing that, you arrive very naturally at the "sharp" and "flat" formulations.