Expanding a bit more on my comment on another answer...
Besides taste of the water itself, the very nature of the water (e.g. minerals and other water factors like pH that was cited by OP) will have an effect on extraction also. That is, different "water" will actually extract differently. This is a marginally distinct from a simple sum-of tastes; i.e., taste-of-water plus taste-of-coffee is not necessarily the whole story.
As I suggested before, try the extreme of bottled water as your comparison: distilled water. Though not always recommended (for a variety of reasonable and/or dubious reasons; more below) it will be the most stark comparison you can make that is reasonably easy. Holding all other factors as constant as possible, brew with your tap water and with distilled water. See if you can taste any difference; I (personally) don't think I could, but I'll certainly try this also! Play around with it, and enjoy the delicious coffee you make while you experiment. :)
There's interesting (to me) debate about the use of distilled water (and, more broadly, "what is the best water to use for brewing coffee?"). To your original question, any water will work for brewing coffee, but there is debate about what is best. Some warn of "dangers" of using distilled water (e.g., metals leaching from the machine, such as in this discussion on Home-Barista) or some quasi-scientific stuff about partial pressures and minerals in this article from The Coffee Brewers.