A Cmaj6 and a Am7 may be the same chord but they resolve differently. 7th's resolve down (usually a step) while 6th's resolve up (again, usually a step). This effects which chord will follow on a basic level. 6th chords will usually resolve to chords a fourth away (C to G) (plagal cadence), and sometimes to a third away (C-E). 7th chords usually resolve to chords a fifth away (A-D)(authentic cadence)., or a second away (A-B).

This introduces common characteristics to these chords as where and how we commonly see them. 6th chords are usually found in chords like ii6 (dorian) which commonly resolve to V and IV6 to I. 7th, well thats easy, V7, which resloves to I, and vii7 resolves to I. V7 can also resolve to VI, a decpetive cadence.

Aside from there close relationship, 6th chords as a part of the harmony are considerable less-common. The applicable and best use (IMHO) is having the "6th" being static. Usually as a pedal. Take a common chord that is seen as a 6th chord, IV6.


Key of C major.
pedal tone: |-(D/?)---|-D-------|-E--------|
chord prog: |-Cmaj ---|-Fmaj----|-(C, Am)-|


here the chord Fmaj can be called Fmaj6 just because the 6th is held long enough to be considered a part of the chord.

* E could imply another possible harmony, Em. I didn't include this one because IV-iii is a relatively weak progression. (IMHO) IV should be considered I in a new key, if you wanted such a movement. The change is from IV to iii, to (IV)I to (V, iii, vii).

Here's another using ii6.


Key of C major.
pedal tone: |-(B/?)---|-B------|-B----|
chord prog: |-Cmaj---|-Dm----|-G7---|


In the second example, the second chord or the third may have a different resolve. B goes to C, so Dm or G7 could resolve back to C maj. Am is another possiblity. Dm could also go to Fmaj. Same with G7, especially if G7 is in first inversion.

This also applies if your looking for 13th chords, although as a much more common practice.

There are chords in the harmony that are 6th chords. Most of the time as a passing chord, on a weak beat within the music. However when you apply them in theory as it is established they are usually better represented as a badly voiced m7 chord, a suspension, or non-harmonic tone.