I have heard that some of the vodkas, which are distilled from potatoes, have a unique flavor.
How are they made?
As I understand it, most (grain based vodkas) are made by producing a sugary liquid (wort) from malt. Yeast is then introduced, and the yeast cells convert the sugars into alcohol-you then distill it, and the result is vodka.
In the case of potatoes, what you have is concentrated starches-how do you convert the starch to sugar?
And why would such potatoe vodka taste different?
http://www.squidoo.com/How-To-Make-Potato-Vodka
Oddly, they leave out the step where you introduce the yeast to actually make alcohol. 
AFAIK, you need a permit to make distilled alcohol in the U.S.
Grain is mostly starch as well. The malting process allows the grain to sprout introducing enzymes that convert the starch in the grain to sugar. I have seen recopies for potato vodka that introduce a small amount of malted grain for the enzymes that convert the starch to vodka. I would think that if you let the potatoes sprout like you do when malting grain you will get the enzymes you need.
http://www.homedistiller.org/forum/viewtopic.php?f=15&t=7974&start=0&st=0&sk=t&sd=a