Is there a reason for that that I’m not aware of?

Is there a reason for that that I’m not aware of?

Either the logic of the harmonic progression of the accompaniment calls for it (it is in any case only another attempt to squeeze music into western classical notation), or it is an indicator where a “blue” note comes from: the Bb may be a flattened B, but not as flat as a Bb on the piano, but something in between. Apply the same to A# and you end up with two really different notes.
Hard to say without seeing the sheet music you are probably refering to, though.
Sonja
The link doesn’t work, but the tunefinder only has one result for “light of mine”, so I’m commenting on that.
As there is no accompaniment, not even chords, I still can only guess. From how I know the song, the Bbs are “blued” versions of the Bs the key signature would call for, and the A#s are there for harmonic reasons.
Actually, when notating unnotatable music like this, the actual way how to do it is often a matter of opinion, and chosing the most readable version.
Sonja
Edit: link works now, thanks.
By the way, this looks like one part of a setting for orchestra or band or something, very hard to play even when you know the song, and not having the complete melody.
Sonja
I took a look at the gif (the link worked fine for me). It appears that someone may have been transcribing what they saw/heard being played. Most of the Bb and a#'s are in places where I would be ornamenting. For example the Bb between two A’s. Another one is a transition between an A and a B note. As has often been noted, the score is often only an approximation of what is being played.
I think Sonja and Lee are both basically correct. Looking at the notation, it appears as if the notes were indeed added for the sake of bluesiness. Technically, in the blues, the natural third is to be used with the I chord, and the flatted third with the IV chord. However, any tone between the second and natural third may be used in either case. You would really have to hear what the transcriber was hearing to figure it out.
I would say the A# and Bb in this case are both “blue” notes, with different designations because the first is between two B’s, and the second between two A’s. Still, I would almost bet if you heard the recording, the A# would actually sound slightly higher than the Bb.
I think BrassBlower is right – these are “blue” notes which means (in this case) lowering the third of the scale a half-step. That suggests that writing the notes as b-flat makes more sence. Also, I think the harmony in bar 5 would probably be a IV7 (C7) chord, which includes a B-flat, not an a# (it comes from lowering the seventh, not raising the sixth).
I suspect the note in bar 3 is written as an a# simply to make it easier to read, since it falls between two b-naturals.
Although theoretically an a# is very slightly higher than a b-flat, I really don’t think that is the issue here — that would be putting too fine a point on the matter, IMO.
–Jay