The distinctions between D-sharp and E-flat only make sense in the context of a key.
For instance, you construct a major key in E-flat and not D-sharp for the practical need to represent the scale nicely on the staff - so each tone in the scale should have a unique place in the staff.
You can construct the E-flat major scale with just three flat tones (Eb F G Ab Bb C D), whereas you would need four sharp and two double-sharp(!) tones if you started with D# (D# E# F## G# A# B# C##). And having to use F## and C## to refer to G and D tones is just ugly.
(I had made a mistake in the earlier version of this comment.)
For instance, you construct a major key in E-flat and not D-sharp for the practical need to represent the scale nicely on the staff - so each tone in the scale should have a unique place in the staff.
You can construct the E-flat major scale with just three flat tones (Eb F G Ab Bb C D), whereas you would need four sharp and two double-sharp(!) tones if you started with D# (D# E# F## G# A# B# C##). And having to use F## and C## to refer to G and D tones is just ugly.
(I had made a mistake in the earlier version of this comment.)