I remember somebody doing an experiment comparing an extract beer brewed with distilled water v. exactly the sam beer brewed with water from the tap. The theory is that a lot of the nutrients will already be in the extract, but if my recollection serves me correctly (this was about 20 years ago) the tasting panel preferred the brew made with tap water saying that the distilled version was OK but tasted blander than the tap version.
Besides city water that you can smell the chlorimine that kills fish I have well water that tastes a little flat. It has a Ph between 7.1 to 7.2 that I use it in the fishpond.
Would well water with it's minerals be better than tap water to brew with?
Very general rule of thumb: if the water tastes good, brew with it.
Distilled water is fine for extract brewing, but you may find that for certain styles (darker or more bitter ones, in particular), that tap or spring water works better for you.
For all grain brewing, distilled water can be somewhat detrimental if it's not treated with an appropriate amount of minerals for the style. Calcium chloride, magnesium sulfate (epsom salts), calcium carbonate (chalk), hydrous calcium sulfate (gypsum), sodium bicarbonate (baking soda), and sodium chloride (table salt) are all possible additives, depending on the initial water chemistry and desired result. See www.howtobrew.com, this site's wiki, BYO back issues, or other many other sources for methods of determining the right ways to affect water chemistry for brewing.
If I'm brewing AG, the minerals go into the strike water before I add it to the mash tun. If I'm brewing extract, I add the minerals at the beginning of the boil, if I add them at all (which I don't always do, since minerals are supposed to be in the extract).
I brewed my first ever batch with distilled water thinking that this would be the best water to brew with. Found out before I even cracked open my first bottle that spring was supposed to be the best and that tap was even better than distilled as long as you could drink it. The brew was an extract brew, but it came out fine. Now I use tap for everything AG or extract, but my tap water is good. If it wasn't I would use spring though not distilled.