Bottled water is sometimes pure tap water itself 
Most of the time, depending on where you live, regualr tap water is perfectly fine thanks to the FDA. The chlorine content and the metals will not do any harm to the Rhac.
Distilled water is basically water that has been evaporated, then the condensation created is transfored into tankes which is then placed into the jugs or bottles or wahtever. It's purpose is to remove impurities from the water, such as bacteria.
I never heard of it damaging cells or anything but I guess it could be possible (as anything is), is there anything else Alan said about this? It sounds like very interesting useful information.
I use tap water, its pretty clean where I live.