Cmaj7 vs. C6 vs. C6/9

Depending on where your musical roots are, you might have stumbled across these three types of chords quite early on. Of course, they are most prominently used in Jazz music but depending on genre and era they are or were very present in orchestral scoring as well.

https://soundcloud.com/robin-hoffmann/cmaj7-c6-c69/s-BBTkrPAE4Va

From a Jazz theoretical standpoint those chords are functionally identical. They all serve as stable I (or IV) chords and could theoretically stand in for each other. 

However, there are situations that indicate the use of one or the other depending on musical context or stylistic approach.

Generally, one could say that major7 chords are the "default" structure for such chord types. Depending on "jazzyness" you might want to add further tension notes to them like 9ths, #11 or 13 but you very often find the basic structure of a major7 chord (root, third and major7) in such voicings.

There is one common problematic situation with such chords though which is when the melody note over this chord hits or sustains on the chord's root note. In these cases you will either end up with a minor second or a minor ninth (+octave(s)) between the major7th of the chord and the melody note. Both cases are often not desirable as they either introduce an unbalancing dissonance to the harmony or obscure the clarity of the melody line by coming too close to it to distinguish it tonally.

In these cases the standard would be to substitute the major7 chord by a 6 chord that would in such cases eliminate these problems. However the 6 chord sounds a bit "simpler" than the already quite simple maj7 chord and can very quickly feel like a "very outdated" sound.

A good alternative in such cases might be the 6/9 chord which when voiced as above has a more modern sound as it includes three stacked perfect fourths that sound a bit more modern and "fresh". If context permits you can even add a major 7th to that voicing as well.

The fact that these chords could stand in for each other also allows to use the maj7,6 and  5 as melodic "cliché lines" over passages where you sustain for longer on one harmony. For instance in a 4 bar passage that is marked as Cmaj7, if the melody on top allows you could harmonize it like 

|Cmaj7|C6|C|C6|

with B, A, G and A as highest note of the chord.

There are a lot of different versions of this top line, some even going in semitones over root, maj7, 7, 6, b6, 5 and back again.