Are diminished fifth and augmented fourth the same intervals? They both consist of six semi-tones but why do they have different names?
In equal temperament tuning, they are the same pitch. So the difference is all in the context in which they are used.
If the music is using a Lydian scale or mode, then it's an augmented fourth. Because it's the fourth degree of the scale.
If the music is using a diminished scale or chord, then the it's a diminished fifth.
In a context like a Blues scale, it's usually considered a flattened fifth. But sometimes it could switch depending on whether the line is ascending or descending.
Every interval has at least two names. Intervals are designated by two factors - the actual names of both notes (with/out sharps/flats), and the number of semitones separating them.
C>F♯ is an aug.4th, whereas C>G♭ is a dim 5th. Yes, they both have the same number of semitones between them but one concerns an F note, the other a G note. Listening to each being played won't give any clue as to what it is, but on paper (sheet music), the way the notes are written will make it clear.
In blues, that interval tends to get called flat 5, whereas in jazz it tends to be sharp 4. But technically it will depend on what the function of that part of the music is, if any.