Although the relative aperture numbers — the ƒ stops — are the same regardless of format, the actual focal lengths of the lenses on small cameras are quite low: 5mm or 6mm at the wide end. That in turn means that the real aperture is small, which means the diffraction limit kicks in sooner, reducing sharpness as one stops down.

The smaller format also means that depth of field is hugely increased even at the widest-available apertures — even wide open at f/2.8, a camera like the Canon G10 has an infinite depth of field if you're focusing farther away than a few feet. So, there's not much difference in that aspect of changing aperture, so from that point of view there's no point even bothering. And that's presumably why there's usually not many choices besides wide open and one closed-down stop like f/8. (Because everything is small, and competitive price pressure significant, adding the mechanics for more intermediate stops is easily deemed not worth it.)

The other aspect of a smaller aperture is, of course, controlling exposure in bright light, without artificially dropping ISO beyond the sensor base or using very high shutter speeds. Some compact cameras actually use a dark neutral-density filter instead of closing down the aperture, specifically to avoid issues with diffraction.