Enharmonics, and why the same sound has two names
Two spellings for the same sound, and why the choice between them is not arbitrary; a note's name carries its function.
C sharp and D flat are the same key. F sharp and G flat too. Two notes that sound alike and are written differently are called enharmonic, and the obvious question is why anyone would bother keeping two names for one thing. The answer is that the name does not describe the sound: it describes what the note is doing.
Where the equivalence shows up
Every black key has two names, one with a sharp and one with a flat. The white ones do too, though less used: B sharp is C, F flat is E, E sharp is F. And intervals inherit the same double identity: from C to F sharp is six semitones, and from C to G flat is six as well.
Augmented fourth and diminished fifth: the same sound, two names, and the difference between them is not pedantry.
The name comes from the scale
A scale uses each letter once. That decides the spelling of every accidental: the D major scale has F sharp, not G flat, because the letter G is already taken by the fourth degree and the letter F needs a note.
The D flat major scale does the opposite: it has G flat, not F sharp, because the letters have to run in order from D flat.
So which name to use depends on the key you are in. In a piece in D major, the black key between F and G is called F sharp; in the same piece transposed to D flat major, it is called G flat. The sound is the same; the name says which scale the note belongs to.
The name carries the direction
There is a second reason, more musical. A sharp, by convention and by tendency, rises: it is a raised note, and the ear expects it to resolve a semitone above. A flat falls. F sharp asks for G; G flat asks for F.
The same key, written two ways, resolves in opposite directions, and the spelling is what warns the reader which direction to expect. The augmented fourth opens outward, to the sixth; the diminished fifth closes inward, to the third. That is why the dominant's tritone is written as a diminished fifth when it resolves to the tonic: the two notes close.
In the chart
The same criterion holds for chords. In a sequence in C major, the passing chord between C and D minor is written C sharp diminished, not D flat diminished, because the bass is rising from C to D and the spelling should say so.
The practical rule is to write whatever keeps the sequence readable: if the bass rises, sharps; if it falls, flats; and, in doubt, whatever the key already uses. A chart with G flat major in the middle of a piece in E major is technically correct and a trap for whoever reads it.
What to take from here
Enharmonic notes sound alike and are written differently because a note's name says which scale it belongs to and where it tends. The scale decides the name by the rule of one letter per degree; direction decides it by the sense of the accidental, sharp upward, flat downward. Choosing the right name does not change the sound: it changes how much the reader understands of what is happening.