Why does wrapping a bottle with a wet cloth produce a cooler bottle than placing the bottle in cold water?

It depends on the variables of temperature and humidity.

Water evaporates from the wet cloth. Water needs energy (heat) to go from liquid to gas and the water gets that energy from it's immediate surroundings - in this case the cloth and the bottle - so the bottle cools. Only if the air is relatively cold and dry, of course. If it's raining, you won't get much cooling from evaporation.

If the bottle is just placed in cold water it is not being cooled due to evaporation and only gets as cold as the water. If the water is flowing and cold, it may readily cool the bottle faster and colder than the wet cloth. If it's just a small bucket of water, it may not cool the bottle much.