The difference between a diminished chord and a half diminished chord lies in the seventh.
- The half diminished chord (m7b5) has a minor seventh (7) - it's a m7 chord wth a flat five (b5).
- The diminished chord has a diminished seventh (dim7, °7). The minor 7th is again decreased by a semitone and hence is equivalent to a sixth (6) - only the theoretical approach is different.
Both are based on a diminished triad. This triad consists of two stacked minor thirds and thus represents a minor triad with a diminished fifth b5* (remember: the minor triad consists of a stack of minor third + major third, which together gives a perfect fifth).
*) You might consider writing mb5 as chord symbol instead of dim or °. However, this is uncommon.
Practically the diminished chord comes hardly ever as a pure triad.
dim (°) usually means dim7 (°7).
Drag the slider under the chord shape to the right to change each half diminished chord to a diminished chord.
The minor 7th of C is a Bb. Thus the diminished 7th is strictly speaking a Bbb. It's up to you if you prefer thinking or writin an A instead.
It's similar with Cb or Fb. In daily life you may sometimes just say "B" or "E".
The diminished 7th chord is completely symetric. It can be seen as a stack of minor thirds. That means that C°7, Eb°7, Gb°7 and A°7 are made of the same tones!
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.