If you play the chord C7, you would play the root, major third, fifth, and minor seventh (C - E - G - B♭), but if you play C6, you play the root, major third, fifth, and major sixth (C - E - G - A). Why does the 7 chord use a minor interval while the 6 chord uses a major interval?
Firstly, C7 should be spelled C - E - G - Bb. C to A# is an augmented six rather than a minor seven, so you need to spell the interval C to Bb (even though they would sound the same, they are written differently).
Secondly, there are different kinds of seventh chords. The one you're talking about is the dominant seventh, which only occurs naturally on the fifth degree of the major scale (the fifth chord is known as the dominant chord of the key, thus the name dominant seventh). So a C7 appears in the key of F major. There are also:
Major seventh chords with a major third and a major seventh (for example Cmaj7 = C - E - G - B).
Minor seventh chords with a minor third and a minor seventh (Cmin7 = C - Eb - G - Bb).
Half-diminished seventh chords with a minor third, diminished fifth and minor seventh (C - Eb - Gb - Bb).
Diminished seventh chord - like the half diminished but with a diminished seventh instead (C - Eb - Gb - Bbb). That last note is read as "B double flat".
In other words, the premise of your question isn't quite correct. There exist seventh chords that use a major seventh. The different kinds of seventh chords arise naturally from the major scale; if you take each note of the scale and stack a series of thirds on top of each one, you'll first get the basic triads, and then different variations of seventh chords. The reason that different intervals appear is that the scale isn't symmetrical. Some notes are separated by a whole step, and other by a half step. Thus, when you move up by a third, you sometimes end up with a major, and sometimes a minor third - which then gives rise to the different qualities of chords.
Edit: As Chai points out in their comment; these seventh chords are written differently in chord charts and lead sheets and such. When it simply says C7, then you're right in that it refers to a dominant seventh chord. The other seventh cords can be written in a few different ways. Wikipedia has a handy chart of common ways to indicate minor/major/diminished sevenths.
I think you should review the basics of interval naming and how jazz harmony chord symbols work.
Those two sources are a decent starting point.
Basically the plain
7 symbol means dominant seventh chord which contains a minor seventh above the root whereas
maj7 means major seventh chord which contains a major seventh above the root.
It's a similar thing with the
6 symbol. It means add a major sixth.
The particular meaning of those symbols will be understood with a review of jazz harmony.
Just make sure you start with an overview of interval naming, because jazz harmony definitions assume those interval names are understood.
As other answers have said, there are many different seventh chords, it's just that the one called 'C7' consists of C major triad and a m7 interval (from I) for the 'seventh' bit. Thus C E G Bb. Being the most commonly used, it's usually called simple 'C7'. There are folk who prefer to call it 'C major, minor seventh', a sort of antithesis to 'C minor major seventh'.
The C6 chord is the same triad, but with a M6 interval from the I for the 'sixth' bit. Thus C E G A.
There is also Cm6, which a little strangely, is C minor triad, but still with M6 on top. Thus C Eb G A. Maybe it could (should?) have a m6 on top, C Eb G Ab, but it just doesn't sound good.
I suggest you don't worry too much about the 'why', just accept that in chord names the intervals are by default major except for 7ths which are minor unless specified as major.
But if you want a historical justification - in functional harmony 7th chords normally had a (primary or secondary) dominant function. You'd find 'cycle of 5ths' progressions, Bm7b5, Em7, Am7, Dm7, G7, C. All the 7ths were minor. Those pleasantly astringent maj7 harmonies (that we're very accustomed to hear now) were less common.
The historical prototype of the seventh chord is the dominant seventh chord in the major mode, built on the fifth scale degree. The seventh above the fifth is the fourth, which is a minor seventh above. The chord therefore comprises the degrees 5-7-2-4, or, for example in C major, G-B-D-F. That is the source of the minor seventh.
The dominant seventh arose first because the nature of the dominant is that of a relatively unstable sonority that leads to a more stable one (the tonic). Dissonance is therefore more tolerable or even desirable in that context, whereas it would be (historically at least) undesirable on the tonic. Major sevenths are therefore quite rare until the last century or two except in dissonant contexts, where they usually resolve according to the normal rules of counterpoint and/or voice leading. In fact, you will (historically) more likely find a major seventh chord as a subdominant (built on the fourth scale degree), where it is also subject to subsequent resolution, than as a tonic.
The dominant seventh chord may be seen as arising from the insertion of a passing tone in the voice that is moving from the root of the dominant chord to the third of the tonic chord (for example, the descending leap from G to E becomes a descending stepwise figure from G to F to E). Alternately, it may arise from the root of a IV chord or the third of a ii chord being suspended into the shift to the V chord (for example, again in C major, in a Dm-G-C progression or an F-G-C progression, one voice has F-F-E).
This is why the unadorned 7 denotes a dominant seventh chord, and other types of seventh chord require some specific modification of the 7 (most commonly maj7 for major seventh chord, where both the third and seventh are major, and m7 for minor seventh, where both the third and the seventh are minor).