If I am in C major I will often see the b6 described as "mode mixture" in my text books or a borrowed note from the natural minor scale. However if I am in the key of C and play a III chord with a major 3rd, this will be notated as a #5. Under which circumstances do we see #5 and under which circumstances do we see b6?
I recommend thinking of at least two issues for this question: the harmonic function/environment of the tone and its resolution.
By "harmonic environment of the tone" I mean its role in the current harmony. If you have a major III chord built on E in the key of C, the major third above E is a G♯ and not an A♭, because a third above E is some kind of G. (The A♭, meanwhile, would be a diminished fourth above E.) Similarly, if you want to play a minor iv in the key of C, your root will be F, meaning the third above must be A♭ and not G♯ (the latter of which would only be a second above F).
As for the tone's resolution, there's a general rule in tonal music that raised notes resolve up and lowered notes resolve down. There are various factors that get in the way of this rule, but it's still largely followed. So if this pitch ends up resolving up to A, it will likely be spelled as G♯; if it's resolving down to G, it will likely be spelled as A♭.
Thus a I–III–IV–I progression will have a G♯ going up to A from III to IV, whereas a I–iv–V–I progression will have an A♭ going down to G from iv to V.
Notate your example and the answer will be apparent. III is a major triad, and will be notated to LOOK like a major triad with a major third (♯5 of the scale), not a flattened fourth. The other commonly altered triad, iv, is a minor chord, so will be written with a minor third.
Think of your other question about diminished 7th chords. Those 4 notes can be spelt in several different ways, and take the harmony in several different directions. Likewise that black note between G and A. Don't worry about what it IS - notice what it DOES.
One of the underlying ideas in "music theory" is that chords are constructed out of thirds. Thus E-G#-B is a chord within the theory, but E-Ab-B is not and would require additional justification to fit within the principles of Tonality and Functional Harmony.
The F minor chord occurs "naturally" in the key of C minor, meaning it's made up entirely of notes from that key. So, when F minor appears within the context of C major, it's considered "borrowed".
E major does not appear in C minor, so it cannot be seen as borrowed from that key.
A b6 would appear in chords arising out of the minor scale: iio, iio7, or ii7b5; iv; VI; or viio7
#5 would appear in chords which require that pitch: I+; III#; or #Vo. However, III# would typically be interpreted as V/vi, and #Vo would most likely be interpreted as viio/vi. Only I+ has a clear meaning within functional harmony.
Where one might see E-Ab-B would be as a chromatic passing tone leading to E minor (iii). E-Ab-B, however, wouldn't be considered a chord in its own right; just an incidental result of horizontal movement.