Brita isn't true RO. It's actually a fine ceramic filter with a charcoal filter layer. It does a great job of filtering out solids and chemicals that taste like crap, and the charcoal layer also grabs some dissolved metals. But it's definately not an RO filter.
If it helps, just think of brita filters as a charcoal and fluff fishtank filter in a neat white plastic shell.
RO filters will stop everything including bacteria and other hazardous contaminants. Brita specifically warns that their filters do nothing to stop stuff that will make you sick, and only "safe" water should be run through brita filters in the first place.
Given the purpose and design behind brita filters, it makes a whole lot of sense to me that vodka run through a brita filter might taste better.
I am not a vodka expert and have never tried it myself so I have no personal experience with this, but it seems to make sense to me.