I know that the A major chord is commonly played by fretting strings 2, 3, and 4, then playing only strings 1 - 5, giving the notes A, E, A, C♯, and E, notes in the A major triad.
If I also add the 6th string, E, that note is also part of the triad. Isn't this still an A major chord? I understand that the 6th string E "roots" the chord at E, but the notes are still the same ones as the triad. Isn't this just a different voicing?
It is a different voicing, but since the lowest note i.e. the bass note is different, it's also a different inversion, and it changes the chord's function a little bit. With the lowest E sounding your chord is an A/E, which is slightly different. It's still an A major, but it's a second inversion A major. (First inversion would have C# as the lowest note. And the regular one with A in the bass is a root position A major chord.)
A/E is commonly used in ending chord sequences like: A/E - E7 - A. When the A/E chord is sounding, I don't feel like being completely home yet, but when the bass moves to A, then I feel completely at home and at rest. Here's a question about the properties of different inversions and the second inversion specifically: Why are second inversion triads considered less consonant than first inversion triads?
Another important thing is that playing both the low A and E at the same time makes the sound a little muddy and unclear. When playing an A/E, you should mute the low A so that there aren't low sounding notes that are too close to each other. Because of the muddiness issue, play either the low A or the E, but not both at the same time. For higher pitches, having notes in a very close voicing is not a problem.
The low E can also be used in alternating bass that's used a lot in country, latin, polka, etc. Play A, A/E, A, A/E ... alternating the bass note between A and E, while keeping the rest of the chord notes the same.
It is simply a different voicing - but it doesn't really sound very good.
The E underneath might just be the 5th of the A chord, but it's just a bit overpowering that loud when it's below the low A.
You can usually get away with it ringing in sympathy a bit, if you're not fully damping it, but hitting it loud & proud doesn't really work.
