Just wondering - when would you really need to know this for practical use?
If I look at C and going up to a B -
a major 7th - how does it help me in thinking of this as a diminished 8th
instead of a CMajor 7th chord - other than a theory test?
If the key is C# minor, then the leading tone would not be C. It would be B#. Say a V-I cadence in C# minor would be G#7 going to C#m. G#7 would have G#, B#, D#, F#.
This choice of key is making it more complicated than it really is. So let's say it's C minor. G7 has a B, not a Cb. Easy enough. So when we have a V-I cadence, you get the note B going to C. If this were, say, a choral piece from England in the 1500s, the cadence would also have this raised B going to C. But maybe another singer has a line that is going down through B and ending on the G of what we would now call the I chord. In that time period, that singer's B might actually be a Bb. Which sounds really dissonant, but that's OK because it's resolving to this beautiful C chord.
In a case like that, you end up with this false relation. It doesn't make any sense in that case to call the B a Cb. It doesn't make any sense to call the Bb an A#. So what you have is a diminished octave or augmented octave or unison.