The refrigerant never leaves the rooftop in one system, and never leaves the chiller plant in the other — the thing that actually travels through the building in a large system is just water.
Every mechanical cooling system runs the same refrigeration cycle underneath — compress, condense, expand, evaporate. What actually differs between direct expansion (DX) and chilled water systems is where that cycle physically happens and what medium carries the cooling effect the rest of the way to the occupied space. Get that one distinction straight and the rest — why homes use one approach and hospitals use the other — falls out almost automatically.