The actual distilling has a huge effect, along with the ingredients (Ciroc is grape, for example and so definitely tastes different than other premium vodkas). I think I have a pretty good taste, but struggle to differentiate based on types of water.

For most vodkas the way it's filtered to remove the less appealing head and tail of the alcohol is important - Sipsmith do a great vodka that isn't filtered at all, they just cut off the beginning and end of every distillation.