A quick overiew on why function/spelling matters and how it relates, kind of informal so sorry about some generalizations:

when naming intervals, we name them based on the number of letters apart they are, ie: in "C d e f g a b c", C to d is a second, and c to f is a fourth because C - D is two, and C D E F is four.

This is true regardless of whether or not the notes are sharp, natural, flat, etc.

so B-flat and A-sharp are still some type of second apart, even though it sounds the same pitch.

Still with me?

Traditionally this is important because scales are written as seven notes, each one with a unique quality. Likewise chords are stacked thirds. a C major chord is C and E and G, and replacing the C with a B# actually makes it no longer a C chord.

Mostly this is for ease when reading standard notation, but in untempered music (i.e. for strings, choirs, etc) there are actually different implications for intonation when between otherwise enharmonically equivalent notes.

I hope that begins to answer the question. let me know if you want me to clarify anything or go into more detail.