You can have either a doubling of aperture and have the power the same, or you can have double the aperture and double the power and have the f/ratio the same.
If you have a doubling of aperture and maintaining the magnification the same, your diffraction disks (Airy disks) are half the width, and the image has 4 times the light coming through. In itself at low magnifications you may not be able to see any more detail as such, but because you now have that much more light to play with you can up the magnification and see more detail, and more contrasted details with the 12" than the 6". You'll note that I did not specify at the same magnification, only that the 12" will outperform the 6" in every way. Having 4 times the light is already performing better by one metric. Having twice the resolving capability is another metric, better utilised with more magnification.
Can you instead show a situation where a 6" scope of any type optically outperforms a well built well cooled 12"?