I bought distilled water from the store for a while and my family hated the taste. How could supposedly pure water taste like that?
Then I bought a distiller and distilled my own water. Everyone in my family likes the taste. What could be different if they're both distilled?
Look at the container. The plastics used for all sorts of bottled water are known sources of toxins. Solvents from the plastics slowly leech into the water. I imagine distilled water would be worse as it has greater capacity to absorb materials.
So, my advice is not to buy water, but buy a distiller. My bet is you'll enjoy the water and keep your hair.