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To me when I play these intervals they sounds the same, yet in my harmony textbooks there seems to be a big difference on how these intervals are treated in voice leading.

The principal difference is that augmented 4ths can (and very often do) move to perfect 4ths, while diminished 5ths are hardly ever allowed to move to perfect 5ths and must be resolved inwards... in fact there are only a couple of times where diminished 5ths are allowed not to resolve.

Why are these two intervals treated so differently?

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    Can you include an example of where an augmented fourth moves to a perfect fourth? I would expect these to resolve more commonly to a major or minor sixth
    – nuggethead
    Commented Aug 18, 2022 at 10:37
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    I too would be curious for an example. Since the two intervals are just inversions of each other, they'll both ultimately want to resolve to the same two tones.
    – Richard
    Commented Aug 18, 2022 at 11:02
  • In Jazz piano, the l.h. will often play this interval (whatever it's called) as ^3 and ^b7 of the chord, descending: parallel, if you like. So there's not much resolution. Just listening, how can you tell what it's called?
    – Tim
    Commented Aug 18, 2022 at 11:07
  • nuggethead in vii°6 progressions and vii7° progressions usually the dimished 5ths resolve while the augmented 4ths don't. By the way, learning vii°6 as a passing chord between I and I6 is one of the first chapters in my harmony book so it must be pretty basic
    – user35708
    Commented Aug 18, 2022 at 11:19
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    I reread the title of the question now. Is the OP asking whether d5 or a4 is more dissonant, or whether/why they are treated differently in voice leading? The two are separate questions
    – nuggethead
    Commented Aug 18, 2022 at 11:37

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You are conflating voice leading with an interval's dissonance/consonance. As others have noted, in twelve tone equal temperament, the two intervals are aurally identical.

Why the voice leading is different between them is a matter of context. Part of the difference is that voices "should" never move to a perfect 5th by similar (which includes parallel) motion. Also, conventionally, the augmented 4th should most often resolve into a 6th, as pointed out by nuggethead, which is the inversion of the diminished 5th resolving into a 3rd.

As a general rule, diminished fifths tend to resolve to a third, and augmented 4ths into a sixth.

In fact, in the first example, on the I6, the alto did not resolve to C# but went to an E only to fill out the harmony, otherwise the chord would be missing the fifth and sound emptier. The E was not chosen because resolving to a fifth is better voice leading!

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The diminished fifth is the inversion of the augmented fourth. The resolution is therefore also inverted. Consider, in C major, one voice has B and the other D. They both resolve to C. This is either a minor third resolving to a unison or a major sixth resolving to an octave.

Similarly, the inversion of a fifth resolving to a third is a fourth resolving to a sixth. You mention that

augmented 4ths can and very often do move to perfect 4ths while diminished 5ths are hardly ever allowed to move to perfect 5ths.

That's because you're not allowed to approach a perfect fifth by direct motion (also known as "similar motion"), but that is allowed for the perfect fourth.

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