As young Mozart said in a letter to his dad, “You tune the sharps flat and the flats sharp”. (*)
The augmented sixth has a characteristic flatted sixth scale step and sharped fourth scale step, in most cases expanding to an octave on the fifth scale step. The tuning of the augmented sixth is considerably different than the tuning of the apparent enharmonic equivalent, the minor seventh.
The brief audio recording introduces the three augmented sixth chords with a C, the tonic; the dominant seventh is introduced by a Db tonic. The German sixth is given with its most common resolution, to a I 6-4; it could also resolve directly to a dominant chord.
And here is a pair of Shenkerian speculations: in Shenker’s terms, the arrival at 2 could not occur on a ii triad because that particular 2 should be tuned low; the correct 2 is tuned high as in a V chord. OR! as in an augmented sixth, even the German sixth, which doesn’t even have a 2 in it.
(*) paraphrased by memory from an English translation of Mozart’s letters, last read over thirty years ago, sitting on the floor of the Sibley library stacks …