How much of an impact does the water (minerals/quality/distilled) have on a beer?
Most of beer is water, so it is vitally important.
Here in Atlanta, we have some of the better tap water compared to many cities, but Monday Night Brewing in particular takes all minerals and chemicals out of the water and adds back in the appropriate properties to match the style of beer that they are trying to brew.
They emulate the water from the origin of the beer style. Just consider that anything but the most pure form of water will impart some taste in your beer good or bad and is worth being aware of.
Water is extremely important. When touring Brooklyn Brewery, the brewers went on and on about how great it is for them to have access to the NYC water supply.
During brewing, the quality of the water is important because the minerals can affect a beer recipe greatly.
It's important enough that many British brewers modify the composition of their water to more closely mimic the mineral content of water drawn from wells near Burton upon Trent, where traditional styles are considered to originate from. This process is called Burtonisation. I've noticed for my part that some traditional bitters I've drunk have had a noticeable and not entirely pleasant aroma somewhat like old eggs, which I would guess is the sulphur content that is added in this process. I don't particularly care for beers that taste like this, but each to their own.
Beer might be completely undrinkable if the water is poor quality. Usually it is ok to use your own tap water, but if it is softened, you'll want to buy bottled water, as the added sodium from the softener will ruin your beer.
I personally use purchased water from the grocery store. Usually 50% distilled, 50% "spring". Though honestly it's probably all tap water from somewhere else.
Water is the primary ingredient in beer. In my opinion, the water quality is more important than anything else. I believe in drinking beer for health, and as I don't drink dirty municipal water, why would I drink beer from it?
Just think about city water from a second. Much of it comes from the same river that the sewage is dumped in. Just because the stink has been taken out of it with carbon filters, does that make it tasty to you?
Also, I don't believe in drinking sodium fluoride. No rat poison in my beer, thank you. I don't care how many people say that it's in there for your smile.
When I buy commercial beer, my primary criteria is the water. If it is fluoridated, no thanks. If it is spring water, super thanks! Spring water is natural and it has natural minerals built in. Also, you will notice that the finest beers have spring water. See BFM of Switzerland and Samuel Smith of England.