Actually it depends on the instrument.
Some instruments can produce different notes for A# and Bb, others can not.
There are different ways to intonate. On one side you have a just or harmonic intonation which is built on harmonics scale (each tone has a a matemathical relation between the base tone), this makes each tonality have its own intonation; on the other side you have temperate intonation which makes a compromise between frequencies and different keys, dividing the interval octave in equally distance semi-tones, to make possible one instrument to play in different keys, always using the same notes.
Here is a good explanation about this. Alsto worth to read this.
In practical terms, to be able to fine tune a chord (just/harmonic intonation in the guitar or different instruments playing/singing together) you must raise or lower some tones. Often the third in the chord needs adjustment. For example the third in F# chord (A#) should be higher than a Bb. If your instrument can't play it (like a piano) you land on tempered intonation, if you can play it (or bend the tone guitar/harmonica/etc) then you can get a just/harmonic intonated chord.
Wheat Williams posted this very clear table on his answer to another question.
Notice how the third in the chord is higher or lower depending on the intonation model you are using. (the A# in my example of the F# major chord).
About the mathematical relation between tones in the harmonic scale: