Hmmmmmmmmmmm....
I did some research. Short answer: Home brewery guidebooks seem to suggest that use of activated carbon (what is inside a brita filter) is strongly recommended.
http://www.alcoholpurification.com/d...distilled.html
I am at a loss, however, to think of a reason why all cheap alcohol manufacturers don't do this--activated carbon is pretty cheap. They may be correct in saying the flavor improves; however, I am still skeptical.
Longer answer: In this situation, the most significant aspect of a Brita 'filter' is the fact that it contains activated carbon. I should note that a brita filter doesn't really filter anything--it's just a bunch of 'pebbles', if you will, and water trickles down past all the activated carbon. If you had a full container of water and you dunked that brita cartridge in, like you would a teabag, maybe swish it around a bit, you should get the same purifying results.
How does activated carbon (AC) work? If you were to blow up a AC 'pebble', it would look like that asteroid with the worm in it, that you saw in Empire Strikes Back. Lots of tunnels, kind of like a sponge. Organic molecules happen past and get stuck inside the tunnels. In addition to this physical 'stickiness' there's a chemical stickiness as well. Since AC is non-polar, it tends to be 'sticky' with other non-polar molecules.
Ethyl alcohol is a pretty polar--if it weren't it would have a hard time mixing with water. So no, it would seem that the alcohol shouldn't get caught in AC.
So what, exactly, would AC adsorb? I did a quick look on the internet for what would cause off-flavors--the best I could find was this article on beer:
http://www.sciencenews.org/articles/20051203/bob8.asp
Assuming that bad flavors in beer = bad flavors in vodka, we have chemicals such as
3-methylbut-2-ene-1-thiol
trans-2-nonenal
furfuryl ethyl ether
Thiol, being a sulfur analog of alcohol, I believe would lead to a polar molecule, so AC wouldn't do much. Trans-2-nonenal sounds like an aldehyde, which is polar as well. Ether is also polar. So none of these would be filterable with AC. I wonder if AC isn't used already for most alcohols, and it's the polar impurities that cause bad alcohol to be bad.