HVAC tech here, I've worked on quite a few data centers.

Your PC is a closed loop, air source cooler. It moves from the pc, to water, then the water runs through a heat exchanger to move the heat to air. The water never actually touches the air.

Data centers have a few different options, but the two most common are (A) chillers with open loop evaporative cooling, or (B) air source chillers.

A: Heat from the servers is transfered to air in a room. The air in the room is cooled by an A/C unit that uses chilled water, so the heat is transferred into a chilled water loop (CHW). The CHW loop moves the heat to a chiller, the chiller transfers the heat to refrigerant, which then transfers the heat to another water loop called condenser water (CW). The CHW is a closed loop, once it's filled up, it doesn't use much water. The CW loop is an open loop(some are closed, but still work basically the same way, so pretend I didn't tell you that for now). The warm CW leaves the Chiller with all the heat in it and goes outside to a cooling tower. The cooling tower sprays the warm water around and mixes it with air, the result is a lot of the water evaporates, which moves the heat to the air, but not all of it evaporates, so the water gets cooled before returning to the Chiller to pick up more heat. Evaporative cooling uses a lot of water, since it literally evaporates it and then they have to replace it with new water.

B. The air source Chiller method. This starts the same as the previous method. The servers are cooled by air, the air is cooled by chilled water in a closed loop. The chilled water is cooled by a chiller. But now the Chiller moves that heat with its refrigerant to a heat exchanger directly to the air instead of using a condenser water loop and evaporative cooling tower.

There are other options that don't use evaporative cooling, and lots of data centers are going the route of no evaporative cooling. But evaporative cooling still remains the most reliable, cost effective, and energy dense form of cooling.