I use a Brita pitcher too, about the only way to make city water palettable! (Grew up in the countryside).
But Brita etc isn't what I think of when I think distilled. MiliQ water at 18.2 MOhm, that stuff is distilled, deionised 'ultrapure' and absolutely rank (granted, not really for drinking though)
The water where I live is great but the plumbing in my house is old and needs to be replaced. For some reason, the water from the kitchen faucet needs to be run for a while so it comes out without rust but the bathroom is always clean. The odd thing is that I replaced everything in the kitchen from the wall to the sink when I remodeled.
Distilled is distilled- there's no other water that's exactly the same, although a good alternative is what comes through a reverse osmosis filter. The taste of the water from my Brita is very neutral and the muni water has a definite chlorine note and nose. It finishes well, though.

