Purified water is the overachiever who uses any method to strip out bad stuff: reverse osmosis, carbon filters, ultraviolet light—whatever it takes. Distilled water is the renaissance purist who only uses one ancient trick: boil it into steam, then condense it back.
Think of it this way: purified water is like a chef using every gadget in the kitchen. Distilled water is like your grandpa using a cast-iron skillet and fire.
Both kick impurities out. But distilled water takes the process to a dramatic, almost theatrical level.