Is there a musical term for two different chords that contain the same notes?
E.g., Bdim7 is B D F A♭ and Ddim7 is D F A♭ B. They both have the same notes but their names are different.
Typically, they are just considered named inversions.
Fully diminished chords and augmented triads like in your example have a very specific concepts tied to the symmetric nature that makes multiple names possible. These chords are known as symmetric chords because they are built using just one kind of interval. If you build a chord by going up in minor 3rds you get a fully diminished 7th and if you build a chord by going up in Major 3rds you get an Augmented triad.
These chords have been informally called enharmonic chords on the site, but that's not a formal term.
There are only three fully diminished chords in regards to actual pitch. They can be spelled different ways according to the key. Your example for Ddim7 is actually spelled incorrectly. It should be D F Ab Cb. The Cb is what shows the quality of the dim7, not a B. That would just be a M6 above D.
Schoenberg called these chords vagrant chords as "such a chord belongs to no key exclusively; rather, it can belong to many, to practically all keys without changing its shape (not even inversion is necessary; it is just to assume relation to a root.) p.195 of his Theory of Harmony.