On keyboard instruments ever since the 17th century, and on fretted instruments since I-don't-know-when, D# and Eb are the same pitch. These instruments have fixed pitches such that, once they are tuned, the pitch remains the same until the next tuning or until they go out of tune. Guitarists can bend the pitch in a way pianists cannot, but still not as much as violinists can.
On non-fretted stringed instruments and some wind instruments, the pitch of an individual note is very strongly under the control of the performer. A slight change of the finger position or a change in embouchure or breathing can alter the pitch dramatically. A good performer will make those changes almost instinctively, either as a result of training or just from listening to what sounds good, and they may not even be aware they are making a change.
In studies of acoustics, tuning, and harmony, the notes of the musical scale are based on mathematical ratios between notes. The notes can sound right while in one key; but when the music modulates into another key, or when a piece is transposed to a different key, the ratios and tuning will be a little "off." This became such an issue with keyboard instruments, as they became more popular, that the concept of "equal temperament" (or tuning) came about. (Remember Bach's "Well-Tempered Clavier"?)
To purists, the distinction between D# (a little lower) and Eb (a little higher) is an important one, and I have seen a lot of discussion of the superiority of "just tuning" over equal tuning; but I must admit, I don't actually like the sound of just tuning on a keyboard instrument, even in music played in the right key. To me, it sounds too much like a calliope.
On a violin, a beginning student will first learn that D# and Eb are the same, but as they advance they will develop more subtlety. When the musician can adjust each note to what sounds right, the difference between the notes can be a good thing. When a musician is locked-in to one tuning (as on a piano), whether it is just tuning or equal or mean tuning (a third type), someone may be unhappy with the results some of the time.
The technical difference between D# and Eb is, as another post says, the difference of a Pythagorean or ditonic comma, about 1/4 of a semitone. For a mathematical explanation, see the Hubs by thunderwave or read "Pythagorean comma" in Wikipedia.
The bottom line, though, is that in music it should *sound* right, even if there is some intentional dissonance.