For example: the G-Mixolydian is G-A-B-C-D-E-F (rather than G-A-B-C-D-E-F#).
Why not just call it C?
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G Mixolydian is a modal scale, more specifically the 5th mode of the major (Ionian) scale. We can't very well call it a "C" scale because (a) G is the tonic, not C and (b) assuming you meant C (Major), it does not have the same interval pattern as a major scale due to the lowered 7th.
As far as its purpose, it can serve many. After all, it's an ordered set of tones, so really it's only limit is your imagination. However, one of its most known uses is as a tool to play over dominant chords. As an example, let's compare G7 (G dominant 7) to a G Mixolydian:
G B D F
G A B C D E F G
As you can see, the G Mixolydian scale is such that the 1st (root), 3rd, 5th, and 7th scale degrees line up with the root, 3rd, 5th, and 7th of a G7 chord. In an improvisational setting (especially when first learning how to improvise), it is useful to be able to quickly refer to such tools as a foundation to use over chords.
I had made a scale sheet for my trombone students a few years ago that I believe is relevant to your question. It's in bass clef (and Bb), but it describes each major mode and its closest identifiable relative chord:
I hope this helps clarify the difference between C Major and G Mixolydian from a functionality perspective.
The G Mixolydian scale only coincides with the key of C major in some systems of tuning, like equal temperament, which is a handy simplification. But it is actually a scale in its own right.
If we point to it and say that "bah, it is just C major from G to G", we are guilty of a form of reductionism, like saying that digitally recorded music is nothing more than a string of 1's and 0's.
The G Mixolydian scale is rooted on G, and the note intervals are understood in relation to G, not in relation to C. If we make the scale pure (just intonation), then we will have a pure third from G to B, and so on. We can write a tune using this scale, which begins and ends on a G chord and uses different harmonic and melodic devices from a G major tune. For example, Since the scale lacks the leading tone (half step approach to the tonic G from below), music which adheres to G Mixolydian will not feature cadences that resolve to G using the leading tone.
Also, it helps to look at the broader world of music. In East Indian music, there are scales that resemble Mixolydian, but in more than one way. There are fine distinctions between different kinds of fourths and thirds and so on. You can hear various Indians scales ("Ragas") on this page: http://www.22shruti.com/index.asp. Just pick a scale in the combo box (e.g. "Khamaj"), and then play the listed notes on the red dots by matching letters.
You can see that the instrument has 22 tones in the octave (called "shruti") from S to S'. The P note is the perfect fifth and M1 is a fourth. Note how there are three notes between these, not only one like in equal temperament.
If you experiment with this, you will see that you can play more than one scale that sounds a lot like Mixolydian. One might be S R1 G1 M1 P D1 n1 S'. But we can alter that in various ways. For example, S R2 G2 M1 P D1 n1 S'. Or S R1 G1 M1 P D2 n2 S'.
The purpose of stating that a musical context is in G mixolydian is to signal to the recipient that the note G is the root of this context. Saying that the context is in C would imply that C was the root of the context.
If the G mixolydian context is a song, or a longer passage of music, then (ignoring temporary modulations and tonicizations) pretty much everything in the song "gravitates" towards the G note and a G(7) tonic(!) chord. The G is "home".
If the G mixolydian context is a chord (i.e. some flavor of the mixolydian scale G7 chord such as G7, G9, and G13), or perhaps a chord progression (such as Dm7-G7), then the notes that will feel like "home" in this context will be the notes of this G7 chord. The C note could be considered an "avoid note" for the down beats.
With a stretch, for the sake of reasoning, you could consider the fact that the G mixolydian mode scale happens to have has the same notes as the C major scale (or C dorian mode scale) as being a coincidence. In the same way you wouldn't refer to a context of A natural minor (or A aeolian) - gravitating towards the A note and Am chord - as being the context of the C major scale. You would say it is the context of A natural minor (or A aeolian).
Consider a song such as James Browns' 'Get Up (I Feel Like Being a) Sex Machine' which is in D mixolydian. You wouldn't say that it is in G major, or that they are playing the G major scale, since the song isn't centered around the note G. It is centered around the note D! And since the seventh note, starting from D, of the applicable scale for the song, (i.e. agreeing with the riffs and the "melody"), is lowered, it is a mixolydian context.
Playing the scale over a C Maj Chord makes it a c Major. Playing it over a G7, for example, gives it the tonality that defines the mixolydian mode. Sweet Home Alabama, Sweet Child of Mine, Norwegian Wood, Hey jude... even the Star Trek theme utilizes the mixolydian mode. So, calling it a c Maj is just wrong. If we used your logic, you telling a musician to play a C mixolydian would lead them to utilize an F maj scale.
See how it attempts to eliminate confusion? And keep in mind... Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian are all modes. Not to be Confused with textbook scales, like the Pentatonic, etc. where the root is the first note. Scales are notes that work in a key and come in two forms. Major and minor. With variations like the Blues Pentatonic with its flat fifth. Using the scale shape and changing the degree on which falls adds interesting nuisances and is why all songs aren't either I-IV -V with notes derived exclusively from the Is major scale in both octave positions or I-iv-v with the minor scale as the only device to which cultivate a melody.
That's my 2 cents anyway. I'm 100% self taught and never been subjected to others views or understanding regarding theory... In a learning capacity at least.
Just know 1-1-1/2-1-1-1-1/2, Ionian through the rarely used Locrian mode in the order I listed, your scales in all positions on the neck in both Maj and min; realize things like
Hey! G Maj scale shapes are the same as Em. So All Maj key scale patterns have a min Counterpart. So a Gm scale can be played over an Amaj. I'll just focus on notes relative to A rather than G. Wow. What if I play a minor scale over a Major chord progression?'
and you will be fine.
Learn it all and you'll naturally gravitate toward your own technique. Whether you fuse modular based ideas together or fall back on the pentatonic (like Hendrix to Page to Jack White and 99% rockers between do) it's all about playing what sounds good and is natural for you to both play and please your ear.