Diminished (dim, °, dim7, °7) and Half Diminished (m7b5, -7b5, ∅)
Big difference in usage
The half diminished chord (m7b5) is a diatonic chord of the major scale. It stands on the 7th scale degree*. That means a Bm7b5 e.g. can be used in a song in C major (respectively A minor).
*) which is the 2nd scale degree in parallel minor.
The diminished chord (dim7, °7) is not part of the major or minor scale.
Half diminished (m7b5)
One of the most common progressions for the m7b5 chord is the 2-5-1 progression in minor vor. Here is an example in A minor (parallel minor of C major). Bm7b5 is the diatonic chord on the 7th scale degree of C major, hence it's on the 2nd scale degree of the parallel minor key A.
E7 is the 5th scale degree of A minor and resolves to the 1st degree Am (that's why we can change Em7 to E7 here).
The diminished chord often acts as a transition chord between the scale degrees of the major/minor scale.