Answer: K.R., as in another previous case let me begin by telling you your email address is invalid. If you have noticed a certain lack of responses to your letters this is the reason; your correspondents do not mean to be unfriendly. Now, for sixth chords:

A "sixth chord," also called the "added sixth chord," in modern terms is a major triad with the interval of a sixth added (as measured from the root of the chord). You've got your root tone, then the third, then a fifth, and finally that extra topping of the sixth. This chord has a sort of jazzy sound and you often hear it as the final chord of a jazz number. The major sixth is the most common one: a major triad with a major sixth:

There's always some confusion of terminology here with regard to traditional music theory. I regard Wikipedia as mistaken in its description of "Sixth Chord." The author of the Wikipedia article says a sixth chord is "any chord or meaningful combination of notes that contains the interval of a sixth." But an ordinary triad in first inversion contains the interval of a sixth, and it is not a "sixth chord:"

In traditional music theory that could be referred to as a "six" chord, but only in the sense that the figured bass for a first inversion triad is "6." There are still only three pitch classes: the root, third, and fifth. Since the third is in the bass the interval of a sixth is created between the bass and the root. But a genuine guaranteed "sixth chord" is one that contains the interval of a sixth above its root.

Perhaps more confusingly, a minor seventh chord uses the same tones as the major sixth chord and if it's in first inversion it is identical with the modern sixth chord. But in classical theory there is really no such thing as a sixth chord, so if that combination of notes appears it's a minor seventh.

So, if you're analyzing a classical piece you won't identify any chord as a sixth chord, though you were certainly find some "six" chords, i.e. first inversion triads, and some minor seventh chords. But if you're writing a modern song you might use a sixth chord, even as a final. Does it make a difference? Of course it does, everything makes a difference. Your purpose in analyzing the classical piece was to understand the composer's intention, and in those days the composer's intention would have been to add a seventh to a minor chord, and the seventh is probably going to "resolve" as sevenths do. In the modern era in pop or jazz the composer was probably actually thinking of a sixth chord, very often with the fourth degree or the tonic as root, and in that case there is no seventh and nothing to resolve; this is just a bit of decoration.