SWMBO and I just got done with our 1st blind tasting of the finished beers from the experiment. The results weren't as definitive as I would have liked, but the tap water beer did have that character that inspired this experiment (although I wouldn't call it harsh). SWMBO preferred the tap water beer, but admitted that her tastes are likely different than most.
The distilled water beer was a smoother beer to me, and was less bitter than I would expect with an IBU:OG ratio of 0.73. It was slightly tart to me, indicating that the pH is too low.
After continuing to finish the two beers from the tasting, the tap water beer became unpalatable while the distilled water beer was still smooth.
While neither of the beers were perfect, I definitely preferred the distilled water beer for its smoothness. I also think that I have correctly identified the quality that I didn't like in the beer that inspired this experiment. If I were to continue to use this extract, I would probably still use distilled water, but add some salts (either chalk or baking soda) to raise the pH.
I still have two of each of the beers from the experiment, so I'll be refrigerating them for another blind tasting with some fellow brewers (probably in 2 weeks+). Sometime after that, I'll be posting an official conclusion to the experiment.