Answer: That throws a lot of music students. Yes, those are exactly the same notes. I'm tempted to take the easy way out and just say that the terminology of pop music conflicts with that of traditional harmony. That would be true, but this could be more interesting than that. Or more tedious, depending on your attitude.
"Sixth" chords have been known a long time, but traditionally they just derived from the practice of figured bass, in which chords were indicated by numbers below a bass line. If you saw a "6" below a bass note it was short for "6/3" and meant that this note was to be harmonized with pitches a 6th and a 3rd above the bass (not counting octaves). That was called a "sixth chord." Musicians before the early 18th century, and many later too, didn't think in terms of chord roots as we do now - CEA and ACE were different chords, not inversions of the same chord. But we finally came to analyze chords by their acoustic root, so that CEA and ACE and EAC were all just different arrangements of the same thing, the minor triad ACE. To know the acoustic root we visualize the chord tones in their closest possible position as a stack of thirds, and the root is the lowest tone.
Add a G to the mix and you have CEGA. Visualized as a stack of thirds we see that A is the root, so in modern harmony we call it an A minor 7 - an A minor triad with a minor seventh added, here just rearranged so that the A is on top. The figured bass abbreviation is A 6/5 (a 7th chord in first inversion). In a way, pop musicians recall the older tradition in calling that a "sixth" chord, regarding the bass C as the chord root. That matters because if the chord is thought to be a C chord it's going to be used where a C chord belongs - like ending a C major piece with a C6 chord, something you hear a lot in jazz and pop. But don't analyze it that way when doing your harmonic analysis of a Bach chorale: for understanding the chord progression those notes form an A minor seventh chord in first inversion: the acoustic root of that collection of pitches is A.
Tired yet? The same pitch, A, could also be called the 13th in a C chord: if you stack thirds high enough you get there: CEGBDFA. But those extended jazz chords are coloristic and get pretty far away from the theory of acoustic roots. And to be called a 13th a chord really needs to throw in at least a 7th, as in C E G Bb A. Chords like the 9th, 11th, and 13th are basically elaborations of a seventh chord - if you don't include the 7th or other higher degrees (degrees past the octave) the A is a mere sixth.
Finally, you could say the name doesn't matter very much. But it's good to have a consistent set of names for things so that people know what you're talking about.
