When looking at bike chains, I usually see silver, gray/brown, and painted options. Is there generally any difference between these in terms of durability, corrosion resistance, or smoothness?
Answer Minimal difference to performance or quality.
The coloured chain tends to be for single speed/fixies or BMXs or bikes without derailleur gears. This is because the front derailleur has to push the chain around, so it rubs. The rear derailleur guides the chain but pushes chain into the sprockets until it has to mount the larger cog. So the pretty colours will get oily or chipped or worn.
So you're more likely to find painted coloured chains in 1/8 inch size, not 3/32 inch and whatever 10/11/12 speed bikes use.