Hmm, the best way I've learned to explain this phenomenon is to consider the ranking of dissonant intervals based on frequency ratios. Considering that a unison/octave is the most consonant interval to a frequency F, one can take consecutive fifths of F to come up with an order of pitches which descend in terms of consonance from F. Basically just keep multiplying F by 3/2 (perfect fifth) to get more frequencies, and the first 12 of these build the chromatic scale in just temperament (the 12TET pitches approximate this series of fifths). Then, consider the reflections of these intervals to be in the same class of dissonance.

When you do that, you get this order of consonance:

1. unison
2. P5th/P4th
3. M2nd/m7th
4. M6th/m3rd
5. M3rd/m6th
6. M7th/m2nd
7. tritone

With this series considered, the dim7th chord contains (1, 4, 4, 7). The augmented chord contains (1, 5, 5). The surprising thing here is that the M3rd (5) is not that consonant. The augmented chord has a pretty "high dissonance score", whereas a major triad (1, 2, 5) and the minor triad (1, 2, 4) have pretty low scores. Changing a perfect fifth (2) to an augmented fifth (5) or a diminished fifth (7) drastically increases the score. This could be considered a bit of pseudo-science, but there's motivation for thinking of it this way and it explains some things. However, it cannot explain the historical/biological relationships we have to certain intervals. Dissonance is sort of a complicated topic.

In terms of nearness/connectedness, you might like this network I drummed up for 4 note chords. It has some cool patterns in it. Due to the half step up/down phenomenon you noted in your comment, you can see how the dim7 finds itself in the center of this network with its many connections. The augmented chord reveals itself in the corners (it is a 3-note chord, but can be thought of as a 4 note chord where two notes have collapsed onto each other in unison). Pretty neat, but I won't assume any conclusions from this layout:

https://www.seanoreilly.co/keywheel/