Richard, the modes ARE different. If you play only the white notes, C major scale, you are probably in the the mode you are in depends on the start note (D=Dorian, A=Aeolian), and a choice of 5 others).
The major key is just one of the seven possible scales/modes.

As a general rule, in neither case should you use a G7 chord, but a G instead, because the G7 contains notes which the ear expects to resolve to a C chord at the end of a phrase. Dominant 7th chords are suited to major and(harmonic) minor modes only, for which they were invented.