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Chromatic transposition, which moves each note by the same number of semitones (e.g. C E G to D F# A) is clearly useful to change the key of a song. I just heard of diatonic transposition, which moves each note by the same number of steps in the scale (e.g. C E G to D F A in C major). However, this must change up a melody or chord progression completely, right?

When is it useful to think in terms of diatonic transposition? I realise that this operation preserves the key, but what does it do and why is it useful? Can it still be considered the same melody or chord progression?

Anna
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    As always, a technique is useful when it leads you to a satisfactory musical result. Dean has given an example of diatonic transposition in the second. There is a fugue (A major for piano) by Shostakovich which has only notes from the tonic chord in its theme, and in the course of the fugue it gets transposed diatonically (scale-preserving) and even chord-preserving (a e a c# e a e turns into c# a c# e a c# a). The result is obviously that all voices are trivially consonant, which is what the composer wanted in this instance. – Kilian Foth Jun 08 '18 at 06:30

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The simple use case for diatonic transposition is fugue & canon. For example, below is the opening of the 6th of the Goldberg Variations which is a canon at the 2nd (i.e the 2nd voice plays the same melody as the 1st, but a 2nd higher). Goldberg Variations. Var 6 - Canon at the 2nd (Czerny ed.) As you can see, the transposition is diatonic, rather then chromatic. One effect of this, as you have deduced, is that the 2nd voice is in a different mode from that of 1st. The upshot of this is that the 2nd voice continues in the same key as the 1st. If you were to play the 2nd voice in the Bach example in the same mode as the 1st (chromatically transposed), the music would modulate every 2 bars as statement (1st voice) caught up with the new key set by the answer (2nd voice).

If this were a fugue, the second voice being "diatonically transposed" would be called a tonal answer as it preserves the key of the piece, whereas if the second voice enters "chromatically transposed", thereby modulating to the key of the second voice's melody, it would be called a real answer, effecting a modulation to the key of the answer.

Dean Ransevycz
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