That's a good question....I would have thought that plants get all the nutrients they needed from the soil and the water was merely used to transport that goodness into the plant....perhaps not......I know that people who keep marine/cichlid fish and use a RO unit to filter the water do have to replace salts in the water or it is to sterile for the fish to live in.....maybe it's the same for plants?
Distilled water contains no plant nutrients at all. Once the plants have used up the nutrients in the soil they will die. Just try giving them a bit of "BabyBio" as well.