Why is a German or Italian augmented sixth chord NOT the same as a dominant seventh chord?

There are two key differences between dominant seventh chords and augmented sixth chords, despite the fact that they sound the same and use the same pitches: function, and spelling. A dominant seventh chord is most commonly used at cadences, and is built on the fifth (dominant) degree of the scale - hence, the progression V⁷-I. In an interrupted cadence it might resolve to chord VI (and outside of strictly 'tonal' music it might encounter different treatments), but it will still be built on V - which gives it the name 'dominant'. In accordance with triadic harmony, a dominant seventh chord should be spelled as a chain of thirds, eg. C, E, G, B-flat (dominant seventh in F major). It does not matter what inversion that chord is then in - it might be in third inversion, where the B-flat is in the bass - one can still explain the chord by spelling it as a chain of thirds. That chord of C, E, G, and B-flat, assuming a perfect cadence, would resolve to a chord of F, A and C in some arrangement.
Above, we have seen that a chord of C, E, G and B-flat is a dominant seventh in F major. Now, let us take the chord of C, E, F-double-sharp and A-sharp. This is an augmented sixth chord in E major - quite remote from F major. Identifying an augmented sixth chord (rather than a dominant seventh) in a piece of music is not always straightforward. The key identifying quality of an augmented sixth chord is, quite simply, the interval of an augmented sixth between its lowest and highest note, when the chord is in root position - in this case, C-natural to A-sharp. By extension, augmented sixths cannot be spelled as chains of thirds. Augmented sixth chords serve a different function to dominant sevenths, in that they prepare a chord of Ic, rather than resolving to chord I. In the case of C, E, F-double-sharp and A-sharp, the voices resolve in a very specific way. The C must fall to a B, the E will stay fixed, the F-double-sharp will rise to a G-sharp, and the A-sharp will rise to a B - giving a chord of B, E, G-sharp, B - E major in second inversion. From here, a typical Ic-V⁷-I progression might ensue (or any other progression leading from Ic). The different spellings of these two chords are crucial in denoting both their functions and the required voice-leadings away from them.
N.B. augmented sixth chords may also appear in different inversions - but the voice-leading will typically be treated in the same way.