The answer is yes and no. The good news is Puerto Vallarta has some of the cleanest water in all of Mexico! The water is treated and purified, but that being said, the distribution system is not perfect. In many restaurants and some rental properties you will get purified water, but not everywhere. Sometimes even if there is a system installed, it isn't always working properly. It is best to ask the staff where you are, and of course when in doubt, go with bottled water! It is very inexpensive and then you know you are safe and won't risk ruining your vacation!
As for brushing your teeth, taking showers and washing your hands definitely do not worry about bottled water for that. It is more than ok for that!
One thing to keep in mind, if you are buying produce from local markets and street vendors, heck even grocery stores, take the time to clean it with more than just water! In Mexico (and other countries!) sanitary practices are not always followed from growing produce all the way through when it makes it into the hands of the sellers and then to you. There is just a lot of opportunities for contaminates, from irrigating with unclean water to fertilizing with manure, you just don't know, so a quick soak it in a few drops of microdyn solution (follow the directions on the bottle) and you'll never regret it!