What is the difference between e flat minor and d sharp major?

For a start, the main difference is between the more martial, joyful major sound and the more sombre melancholy minor. The fact that the first note of each key (the tonic) is the same (Eb is the same as D#) is probably the only similarity. Thereafter the flattened third and sixth note of Eb minor will distinguish it in sound to the scale of D# major. The second difference is the writing; the same notes, while sounding identical, will look different on the sheet music.