Well, without any further context there is no possible distinction between a minor third and an augmented second as they are indeed the same note, technically.
However, the phrases minor third and augmented second make reference not only to that space of three semitones, but also to the relationship that this interval plays within a given chord or scale. Since almost every scale known to the Western world has some form of third, a scale or chord with an augmented second would most probably have another kind of third as well, making the augmented second seem like an added dissonance in context.
Consider this chord: C7#9. (The #9 can be considered equivalent to an augmented second). The naming convention of the chord assumes there will be a major third. So the chord is 1, 3, 5, 7 and #9. The last note is not an essential part of the chord's structure, and will sound dissonant.
If you just have a C and a D#/Eb, it doesn't really matter whether you call it a D# or an Eb. But when you're talking about the role that note plays within another structure, then there are reasons for naming it an augmented second (to make clear it is not a minor third).
Hope this helps!