Answer: The thing is, no one ever sat down and decided how this was going to work. It just developed organically. When the musical universe began there were no sharps or flats - just a pattern of steps that, to simplify greatly, ended up being represented as what we now call the natural minor scale: A B C D E F G. Nothing else. There were seven notes, repeating at the octave, and seven letters did the job.

But then, again simplifying, musicians wanted to sing the known patterns starting on a different note, and that required adding the first "black key," B flat. By lowering the B, one could sing, starting on F, the same pattern that started on C. You needed a lower form of B, and it was called "soft" B, B molle, the origin of the flat symbol. It was not a higher form of A, because you sing the notes in alphabetical order.

You can see where this is going. Eventually it was desired to sing the same pattern of whole steps and half steps beginning on any note, and that meant putting a half step between each whole step of the keyboard. But whether it was called a flat or a sharp depended on the context: if a note is raised in pitch it is sharped, and that is not the same thing as lowering the next note up: the two are in different scales and imply different harmony.

That's why our current keyboard is designed to show what are now called the major and natural minor scales on the white keys, and the black keys fill in where needed. No one ever designed it, but personally I think this system works nicely, because its history shows in its terminology, and the essentials of harmony are reflected in those names. Also, the resulting 2,3 pattern of the black keys gives you some valuable landmarks when placing your hands on the piano.

Back in the 80's there was a music program, possibly the first personal software for music notation, called MusicWorks. The programmer had probably never been a music student and had the idea that one could get along without flats, because after all, aren't flats and sharps the same thing basically? So MusicWorks had no flats. Just sharps. Of course that programmer was deeply mistaken and this pretty much ruined the program, which otherwise was very welcome. While a student I used to transcribe old music notation with MusicWorks and had to improvise with a pen when flats were needed because you simply cannot write A, A#, C, when what you mean is A, Bb, C. The musical meaning is different. It's worse than misspelling a word; it changes the meaning.

There's more on this topic in questions 9, 31, 65, 104, and 122.