Camouflage on airplanes helps make them harder to see - blue or grey hides them from the ground or lower aircraft, brown/green/blue hides them from above or when on the ground. The actual colours and patterns used vary depending on local conditions (e.g. deep blue undersides are better in the tropics, grey in northern Europe) and amount of effort that air forces were willing to put into maintaining complex paint patterns. However, the paint slows the aircraft down, due to a rougher surface than bare metal (smooth paint is shiny and not much better than no paint for camouflage), and adds weight.
With early aircraft, this didn’t matter much, you HAD to paint them to protect the wood, steel and fabric structure from the elements, so using a camouflage colour instead of some other colour was no big deal. By the last year of WW2, however, aluminum was the norm for aircraft structures, and didn’t need (much) protection. Throw in a tactical situation where it didn’t matter whether the other guys saw you or not, since the Allies’ near-total air superiority kept them from doing much about it, and the disadvantages of painting the aircraft outweighed the advantages and they stopped doing it.
The US started painting planes again when the tactical situation changed to make it worthwhile again. The development of better anti-aircraft weapons allied to the recognition of the need for more “close to the ground” kinds of missions was one factor.
Naval aircraft are almost always painted, to help control the corrosion problems caused by operating in a salt water environment.
Airlines paint their planes for advertising and customer appeal, factors which significantly outweigh the disadvantages from their point of view.