Every intercooler has to dump heat into something. Air-to-air systems dump it straight into the air flowing past the car. Air-to-water systems dump it first into coolant, and then the coolant gives it up through a separate radiator or a tank.
Both start from the same problem, hot compressed air leaving the turbo, and both end by rejecting that heat to atmosphere. The difference is the path the heat takes and how much hardware that path needs.
Air-to-air is the simpler of the two. Charge air passes through the core, ambient air passes through the other side, and the heat moves from one to the other through the metal. There is no coolant, no pump and no second radiator to fail.
Its weakness is that it depends on airflow. At speed it works well; in traffic or during repeated standing starts it relies on whatever the car's fans and motion provide, which may not be much.
Air-to-water puts a liquid between the charge air and the atmosphere. Coolant passes through channels next to the charge air, absorbs heat and is then circulated to a heat exchanger where that heat is released. The liquid carries heat away from the core faster than air can.
That lets the core itself be more compact, which is often the point. The trade is that the heat still has to go somewhere, so the system needs a pump, a circuit and usually a radiator or an ice reservoir to be complete.
Air-to-air suits road cars, track cars and anything with steady airflow and space at the front. It is robust, simple to run and has little to go wrong. It is the default choice for most builds.
Air-to-water suits layouts where a front-mounted core will not fit, or where charge pipes would otherwise become long and complicated. It also suits short, intense events where the coolant's capacity can absorb a lot of heat in a brief window.
Air-to-water frees up space at the front of the car because the main core can sit closer to the engine, or on top of it. That can shorten the charge path substantially, which helps response. It also makes a compact intercooler possible where a large air-to-air unit simply would not fit.
The cost is more parts. A pump, hoses, a heat exchanger and often a reservoir all have to be packaged and kept working. Each one is another potential leak or failure point, and the system needs bleeding properly to function.
The deciding question is how long the car has to stay cool. Brief bursts reward the thermal capacity of a water system; long sustained running rewards the airflow-dependent but endlessly self-recovering air-to-air core, as long as it gets clean air.
Water systems also need somewhere to store and manage the coolant, and a well-made aluminum water tank is part of that. Match the system to how the car is used, not to which one sounds more advanced.