On the Pride last summer the ship supplied a HUGE bottle of Evian in our state room that was replaced when empty. The water however in the dining rooms is not bottled but it was fine. You can indeed taste the difference (taste like tap at home). When on deck I took the bottled water from our cabin and our stewart always replaced it with a new one.
water on the ship is fine - watch out for the water in ports. you can fill a sport bottle with ice and water from the lido deck and take it with you or buy bottled water in the ports (cheaper than on the ship).
What they do on the ship is boil the seawater and catch the condensation for the drinking and bathing water. I think it is called distilling the water but I'm not sure on the technical term. It is purer then bottled water. We just bring along a sports bottle that we bought on our first cruise and fill it with water and ice for bringing to the ports otherwise the water from the tap is fine, but I drink tap water at home too.
There are some sports bottles, I think Brita makes one, that have a filter built right in if you have any concerns about the water. I have seen them at Walmart are they aren't too expensive if you are interested in something like that.
On our last cruise (Royal Caribbean-Voyager ) we were able to take on board 2 cases of bottle water. What you should do is call the cruise line and ask if you can do this. It did save us money - since we are a family of 7.