Purified could be anything, most likely a filtering of some type.
Distilled is water that is boiled, then condensed. It's the most pure water you can get.
For something like a radiator or lead battery, distilled is the only way to go. Purified is drinking water.
We human being take in more than just the water we drink. Even if you drink only distilled / RO / DI water, you still eat food with lots of minerals, and the body will keep them in a balance.
Unless you have a kidney problem.
Distilled water is great for drinking. Nothing wrong with it and very easily absorbed. A lot of athletes will drink distilled water. I do not understand why people think its bad to drink.
The reason some think it's bad is because IF you didn't brush your teeth, and IF you didn't eat well, you wouldn't get those needed nutrients and fluoride that comes in tap water that could help you out.
The internet is a great medium for kooks to spread junk science. Sorry, doc, I am a chemist, and the difference in distilled and tap water isn't enough to make near the difference you suggest. Tap water is seldom anyways near saturated with anything and will adsorb about as much as regular tap water.