The difference in a lot of the keys is subtle grammatical/spelling differences in the theory of how the music is written, and the language you use to describe the notes and chords in the piece.
A# minor is the relative minor of C# major, and has 7 sharps in the key signature.
Everything would resolve to an A# (needing a leading tone of G double sharp… yuck) and the chords would be identified from the key signature (all sharps).
Bb minor is the relative minor of Db major and has 5 flats in the key signature.
The leading tone is an A natural (much nicer than G double sharp) and all the chords would be identified inside of the 5 flat key signature.
It would sound exactly the same.
Why one instead of the other?
Depends on the instruments you’re writing for, what the composer feels about key signatures (Beethoven believed different keys communicated some feelings better than others), stuff like that.