Question: In a song I am learning, there is a section of the song that has the chord progression G#m | A#m7b5 | B6
I sometimes get the fingerings wrong and use dimished 7th chords on guitar instead of m7b5 chords and in this case I actually think the dimished7th chord sounds good too. So I wanted to ask, what is the difference functionally between the two? They both seem to work in most situations as leading tone chords to a tonic chord one semitone above (B in this case).

Answer: 
The tritone A♯ and E resolves nicely to B and D♯. Other notes in the chords are less important.

That tritone is found in F♯7 (the standard dominant of B major). It's also found in A♯m7♭5 and in A♯dim7. (Also in C7, if we want to play with ♭5 substitutions.) So they'll all resolve in a nice functionally harmonic way to B major.

There are other ways to approach B major. You can 'plane' up from A♯ major or A major. Don't struggle to analyse this as functional - being the same chord shape and being adjacent are sufficient.

So does this mean a major triad, a dom7 chord, a dim7 and a half-dim7 all 'function the same'? No, that's far too general a statement.
While m7♭5 chords and dim7 chords are fairly interchangeable as dominant-function chords (don't interchange them if the melody note is the 7th), the dim7 chord is a much better **pivot chord** between keys than the m7♭5 chord, as any given dim7 chord is an enharmonic re-spelling of 3 other dim7 chords, while a m7♭5 chord cannot be enharmonically re-spelled into any other m7♭5 chords. When a dim7 chord is acting as a pivot chord, interchanging it is unwise.