Why should I add water in cooler but not in air conditioner?
Presumably when you say "cooler" you mean an evaporative cooler, sometimes called a swamp cooler.  These work in dry climates by evaporating water, which cools the air passing through (and also increases the humidity).  It doesn't actually remove heat - just uses some of the heat to evaporate water, which reduces the temperature of the air. As the water is evaporated, it has to be replaced.

An air conditioner works by using a thermodynamic refrigeration cycle to remove heat from the air.  As the air is cooled, humidity in the air will actually condense so an air conditioner discharges water.