In a previous question I made an attempt to describe how well a pair of binoculars would perform. I did so in the context of the observer being able to see 'point sources' such as stars, but did not consider the contrast between these stars and extended background emission such as the 'sky' or faint nebulae. For point sources, it is true that the brightness of the unresolved star depends on the light gathering 'power' of the binoculars. The larger the area of the aperture/lens of the binoculars, the fainter the star you will be able to see in principle provided that the background you are seeing it against is perfectly black. There is, of course, a second component to this and that is that, except when you are far from city lights, you are almost always in practice concerned with viewing a star against a bright background.
Although for point sources the brightness scales with the area of the lens relative to the area of the human eye's pupil, for extended emission, the scaling is a bit more complicated. For point sources like stars, the arriving photons are spread out over the size of the 'diffraction disk' of the instrument which is set by the diameter of the lens. Provided you are not over magnifying the star images, all of the photons from a single star arrive within this diffraction disk, and so long as you cannot resolve this disk, the photons all seem to arrive at infinitessimal 'spots' in the field of view. Increasing the aperture size increases the number of photons collected per second and only serves to increase the brightness of these infinitessimal resolution elements in the field of view. The brightness of the stars then scales directly with the area of the binocular lens.
For extended sources such as the sky and nebulae, we can resolve them in the field of view. Now the relationship becomes more complicated because, the sky or a nebula emit a fixed number of photons per square degree. For instance, suppose that we had a hypothetical pair of binoculars which had a field of view of exactly 1 square degree at a magnification of exactly 1.00x. The sky emitted 6400 photons into this region per second. Now suppose we increased the magnification by a factor of 8 to 8x. The field of view is now 1/8 as large as it was at 1x and we are only seeing 1/64 the original sky area and therefore 1/64 the number of photons per second. The sky will appear 1/64 as bright as it did at 1x, and each second the field we see at 8x will be emitting 6400/64 = 100 photons per second. This shows that at a fixed aperture, extended sources such as the sky and nebulae will appear fainter as you increase the magnification PROVIDED that they fill the entire field of view at all times.
To ask how well you will see stars against a fixed diffuse background, you now get to worry about the magnification and the aperture. The aperture will fix the number of photons you detect from unresolved stars, provided that the magnification you use does not over-resolve the light from the stars. You now see that if you want to observe faint stars against an urban, illuminated sky background, you can generally obtain some advantage by observing with higher magnification. There is an important caveat.
If your sky background illumination per diffraction disk of your instrument equals the amount of light coming from a star of a magnitude, m, no further magnification will allow you to detect the star against this background because it contributes just as many photons per diffraction-limited disk as the background. For the naked eye, you can estimate the magnitude where this limit is reached by seeing what star is the faintest one you can just make out. It's magnitude limit is set by this equality based on the size of your pupil as the aperture defining the relevant 'airy disk'.
For more on this, see the Article by Alan MacRobert in Sky and Telescope, May 1995, page 48.