A sharp and B flat name the same equal-tempered pitch, the last of the five enharmonic chromatic pairs in this atlas. B flat is by a considerable margin the more commonly seen spelling in practice, since B flat major is a foundational key for the transposing brass and woodwind instruments standard in concert bands, most of which are pitched in B flat themselves, making the key and its written pitch A sharp's enharmonic twin one of the most frequently encountered in band repertoire specifically, even though it appears comparatively less often in string-dominated orchestral writing.
The prevalence of B flat instruments in wind band tradition, trumpets and clarinets prominent among them, is a matter of instrument manufacturing history and acoustics, tracing back to the natural harmonic series available on early valveless brass instruments, and has no connection to any correspondence system; it simply means that a great deal of concert band music is written and read in keys built around this pitch.
This same enharmonic ambiguity, one sounding pitch with two written names, runs through the whole chromatic scale under equal temperament, and which name a composer or arranger chooses generally follows the surrounding key rather than any fixed rule tied to the pitch itself. A brass or reed instrument built in B flat sounds a written C as a concert B flat, which is why band scores are conventionally written in keys convenient for those instruments even when the underlying concert pitch would be notated differently for a C instrument such as a piano or violin.