I've been thinking about a simpler way to explain it. It also explains why you can have double flats and double sharps (leading to 35 possible note names for 12 different pitches). Here goes...

the vast majority of western music involves 12 notes in an octave[br]the vast majority of western music is based around a scale consisting of 7 of those notes specific to the choice of key (the notes are called the diatonic notes for that key)[br]a particular note in a piece is functioning either as a diatonic note or as a note a semitone higher or lower than a diatonic note[br]when expressing a note that is functioning as a raised or lowered note, you use the same letter name as the diatonic note you are raising or lowering. e.g. a raised G is G♯ and a lowered G is G♭.[br]if the diatonic note is already written with a sharp, the raised note has a double sharp and the lowered note has a natural symbol[br]if the diatonic note is already written with a flat, the raised note has a natural and the lowered note has a double flat[br]but in all cases, the letter part of the note name stays the same[br]So, imagine you're in the key of Gm. The diatonic notes are: G A B♭ C D E♭ F. What does A♯ mean? It means you've taken the second note of the scale and raised it. What does B♭ mean? It means the third note of the scale.

In 12-tone equal temperament, they may sound the same;you may play them the same on the piano or the guitar. But if the function of the note at a particular point in the piece is as the third note in the Gm scale, you can only write it B♭ and not A♯. A♯ means something completely different.

It's the musical equivalent of "hear" versus "here". Just because they are homophonic doesn't mean they are the same word. Similarly, in western tonal music B♭ doesn't mean the same as A♯.