Answer: They're the same pitch on a fixed-pitch "equal-tempered" instrument like the piano but not necessarily the same pitch on instruments with free intonation, like the violin. Perhaps more importantly, the two different notes have different implications for the harmony. If the reader is to understand that this is meant to state the c minor harmony then the middle note must be Eb because that creates a visual triad. A D# would imply that you are not actually intending a c minor harmony but are attempting to do something else, perhaps moving though some kind of modulation to a different key. The music becomes harder to understand and is also harder to read; it would be comparable to mispelling words arbitrarily when writing a story. Notes that sound the same on the piano but are "spelled differently" in notation are known as enharmonic equivalents. The only purpose of notation is to convey your meaning, and the choice of enharmonic is part of the meaning.

Melodically there are differences, too: because a leap from C to D# is an "augmented second" a player will instinctively think that this is being treated as leading to E natural. A violinist might raise the D# a little extra to enhance that. But a C to Eb doesn't carry that implication of upward movement, and there is no temptation to stretch the pitch. One would expect an Eb after a C to either to be stable or else move to D. These are subtle things, but worth observing because they enhance the meaning of the music.