So it's an air conditioner, right? Technically, yes -- it conditions the air by cooling it, but it works much differently than what we commonly think of as an air conditioner.
Standard AC units work by passing air over a set of coils filled with a refrigerant like Freon (a trade name for a variety of chemical blends), which heats and cools as it's compressed and expands. The air is cooled by the coils, sent into your house, and then re-circulated over and over through the machine, venting hot air generated in the process to the outside. (For more about standard air conditioners, see How Air Conditioners Work). It's a closed process -- leaving a window or door open allows the cool air to escape and makes the air conditioner work harder to supply cold air.
Swamp coolers are an open system. They rely on the flow of air through the building to direct the cool air, and since they always need hot, dry air to evaporate the water, it needs to displace the air already in the house. Both systems can use either a large central unit or small window units, but air from the swamp cooler needs a way out. Opening and closing windows and doors controls the air flow from the swamp cooler to different parts of the house, while central air conditioners use ducts to direct the flow. Swamp coolers can also use ducts in some cases, but they need to be larger than traditional air conditioner ducts to account for a greater flow of air from the swamp cooler.
Standard air conditioners also dry the air, condensing water vapor from the cooled room as it passes over the cold coils. The water drains outside -- that's the distinctive drip you'll feel if you stand under a window air conditioning unit long enough. The result is a dryer room, and in humid climates, that can be a good thing. Too much humidity can prevent perspiration, which is how we cool ourselves naturally. Since swamp coolers work by putting water into the dry air, they act as humidifiers. This is great in dryer climates, because humidity can also be too low for comfort. Under the right conditions the water-laden breeze also can have a secondary effect of helping the skin's perspiration, resulting in an even cooler feel than the swamp cooler would give on its own.
Because of the different ways they work, you can't run a swamp cooler and a standard air conditioner in the same house. They would cancel each other out, just like running a dehumidifier and a humidifier in the same room.
So, if you can only pick one, which is better? The swamp cooler or the air conditioner?