An evaporator coil is where a liquid changes state to a gas, while a heat exchanger is used to move heat from one substance to another. An evaporator coil is a type of heat exchanger.
One place evaporator coils can be found is in air conditioners. When a low-pressure liquid is routed through an evaporator coil, it changes its state to gas. When this change occurs, heat is absorbed by the liquid. In the case of an air conditioner, heat is pulled from the surrounding air, making it feel cooler. The evaporator coil is acting as a heat exchanger by moving heat from the air to the liquid inside the evaporator coil.
Not all heat exchangers use evaporator coils, however. For example, a gas furnace uses a heat exchanger to warm the air before it is circulated through the house. In this case, a gas burner creates hot combustion gases that move through a series of metal tubes. Air is pumped around the tubes and picks up the heat from the combustion gases. In some furnaces, the combustion gases are then sent through another heat exchanger to extract even more heat.
Unlike an evaporator coil, at no time does the air in a gas furnace heat exchanger change its state. Many different types of heat exchangers exist depending on the application and how heat needs to be managed.