there is ONE solution that hasnt been mentioned.
that being evaporative cooling.
a large, noisy, but quite effective means of subambient cooling without a peltier.
it operates on the same principle as your body sweating. the highest energy molecules will evaporate out of the system, resulting in a lower overall energy concentration. that evaporative process to create water vapor requires a ton of energy from the system; reducing the temperature of the sourrounding water and air, but increasing the humidity of the air.
the drawbacks, for best results, your talking about literally using a shower head raining water down a 10" diameter PVC tube about 6 feet tall, with fans forcing air up and against the water flow. then a seperate loop(as the open loop will get dirty quickly from the dust mixxed with it, plus water stabilizers would be impracticle in an open loop) so this is going to be large, loud, and expensive.
guess Ive kinda shown why no one uses it...
but you asked!
PS. as has been said, water and air are 2 means of the same thing, all of them pull ALL of the heat into the air. water may absorb more heat during operation, might mean itll be 30sec-1min for the exhaust to get hot, but all of that heat will just dissipate in the minutes after the system is turned off till it comes back to ambient.
the heat in your room is a direct result of the heat put off by the PC components.
infact, a water cooled will be worse, the pump itself generates a significant amount of heat, as does the friction of the fluid in the loop. hardly an insignificant amount either. but it falls under the same lines as a peltier creating heat, its an acceptable flaw in that the radiator of a water system can effectively dissipate all of that heat, and plus the system heat, with ease. but the ROOM will be all the hotter for it.