We wouldn't have to purify water if there weren't so many contaminants purposefully added to tap water.
For example: fluoride - it is completely unnecessary but is justified by saying it makes teeth stronger. Not only is this a bogus claim, but I have documents from three studies showing that fluoride reduces the IQ in children as their brains develop by up to 6.7 notches on the IQ scale. It has other side effects like dental fluorosis. And that is just one contaminant.
I use an "under the sink" reverse osmosis filtration system (~ $150 at Lowe's) for all my drinking and cooking water. It removes 99.7% of fluoride and similar amounts of other contaminants. OK . . . it also removes some good things, but I can get those in other ways, like unprocessed sea salt.
There is also the question of water absorption while showering/bathing. I've read where the average person absorbs 2 cups of water through their skin every time they shower. If the shower water is not filtered, everything from the tap water is brought into our systems.
Distilled water is very good, but requires a lot of power to produce. New systems are becoming widely available for pulling water out of the humidity in air.
Water in bottles can contain anything. I find it amusing to see the water vending machines, as they simply filter tap water.
While in Mexico, I visited the "Ozona" water company and got a tour of their facilities. They boasted their "Ozone Purified Water" until I asked why their ozone generator was unplugged.
Water is so important to our health - choose your source carefully.