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!
music.stackexchange.com/users/12266/Grey
A key thing to keep in mind is that technically a minor 3rd and an augmented 2nd are different pitches (have different notional fundamental frequencies), at least in anything other than equal temperament.
In just intonation, these two pitches differ by approximately 40 cents ( list of intervals ), enough to make a perceptable difference in the degree of consonance.
(Also note that some types of mean-tone temperament can also represent this difference).
Thus these different notes are different and have different harmonic behaviour in the context of a chord.
Even in the context of music intended for performance in ET, notating the enharmonic notes in particular ways can provide the performer with information on the composer's intent.
music.stackexchange.com/users/2639/Dave
One reason is that if you're specifying an augmented 2, its probably because you have an augmented second and a major third in the chord.
These notes are only a half step apart, and that is very dissonant.
Since we don't have, for example, a "minor fourth" interval, the third will always be major if an augmented 2 is involved.
I guess there could be exceptions, but if there isn't also major third in the chord, it would just be written as a minor third interval.
music.stackexchange.com/users/12118/Eichhörnchen
It seems to stem not from the meaning that we attach to the words now, but the past.
Consonant meant it sat well in the key, dissonant, the opposite.
So, when WRITTEN in music, a minor 3rd belongs in a given set of notes, whereas a #2 is not found.
It appears to be more of a technicality than a reflection of what it actually sounds like.
Turn a minor 3 upside down, and it's a major 6th.
Supposedly consonant.
Turn a #2 upside down, and it's a diminished 7th.
Both augmented and diminished intervals were labelled dissonant.
music.stackexchange.com/users/2808/Tim
A sound example would make it much easier.
Try this experiment:Play a Cmajor scale and some I-IV-V-I in this key, then suddenly play the harmonic interval C-D#.
It will sound dissonant in this context, the D# sounds like a leading tone to E (your brain will ask for solving it to E).
You don't need to have the C-E in the same chord.
It is enough to have the key context in your mind-ear.
Then play a Cminor scale and some cadences in this key, suddenly play the C-Eb (actually same piano keys as C-D#).
In this context they will sound consonant, that is the reason for different name.
So they are different both in the writing and in the sound, in a tonal context.
They only sound the same in atonal music.
You would never find Beethoven mistaken D# for Eb in his written scores, but I wouldn't say the same for some common chord charts you find around.
music.stackexchange.com/users/43100/Prox
I am currently reading the Jazzology and on the chapter per the consonant and dissonant intervals I came across this statement:   while a minor 3rd is consonant, an augmented 2nd is dissonant.    (page 6)   So my question is this:  Why is a minor 3rd consonant but an augmented 2nd dissonant, since they --technically-- are the same note?
Why is a minor 3rd consonant but an augmented 2nd dissonant?
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<theory><intervals>
music.stackexchange.com/questions/21910
music.stackexchange.com/users/7356/Shevliaskovic
A key thing to keep in mind is that  technically  a minor 3rd and an augmented 2nd are different pitches (have different notional fundamental frequencies), at least in anything other than equal temperament.  In just intonation, these two pitches differ by approximately 40 cents ( list of intervals ), enough to make a perceptable difference in the degree of consonance. One reason is that if you're specifying an augmented 2, its probably because you have an augmented second  and  a major third in the chord. Without further context, it's impossible to give a precise answer to this question. However, in just intonation, an augmented 2nd and a minor 3rd are in fact not the same note. Additionally, an augmented 2nd may be noted as such because it appears at the same time as a major 3rd, in which case, it would be dissonant.
It is difficult to provide a precise answer to this question without further context. In just intonation, these two notes are actually not the same note, though they are in equal temperament. One possible reason could be because the augmented 2nd could appear at the same time as a major 3rd.