I think the answers above pretty much answer the question, but I needed to kill some time and typed up my own thoughts.
I won't pretend to know anything about making vodka except what I just read online:
http://www.essortment.com/food/makevodkahome_sbcy.htm
Looking at the method, it seems that pure vodka is intended to be simply a 40% aqueous dilution of ethanol. I would guess what separates the different brands is how close to the intention they actually get.
I do know a little bit about distillation from my chemistry background that might be helpful here. Essentially, vodka is made by fermenting a mash of carbohydrates and yeast followed by distillation and purification of the ethanol from that mash. This contrasts to many other alcoholic drinks where the mixture is usually filtered to remove solids allowing all the natural flavors from the mash to remain.
Distillation is simply a method of purification that boils a mixture and allows you to separate components out of the mixture based on their boiling point. You slowly increase the temperature and the lower boiling components usually come off first, followed by higher boiling points. So, if you want to distill ethanol with a boiling point of 78.4C, you could theoretically boil the mixture at ~77.5C to remove any lower boiling compounds such as methanol (the part that causes blindness, liver damage, and death!) and then increase the temperature to 79C to boil off the ethanol into a separate collector.
If you've noticed I've used a lot of conditions like "intentions" and theoretically", it is on purpose. In reality, there are many complications that lead to gray areas. As an example, some liquids, when mixed in the right amounts will form something called an
azeotrope where both compounds will boil at a lower temperature than either would individually. Additionally, if the boiling process is too fast, you can have a range of temperatures in your boiling vessel and fast moving convection currents that can drive vapors and aerosols with multiple boiling points quickly over to your collector at the same time. This is why instructions say you should distill at least 3 times, always discarding the first and last bits you collect. Distillation is not perfect, but each time you distill, you increase your final purity (although azeotropes can put a stop to that if you have any in your system). As a final purification, it is commonly filtered through activated charcoal which is an excellent absorber of a variety of organic compounds that commonly cause odors and colors. The final step is the addition of a pure water to dilute the pure ethanol back to 40%.
So it seems that the deciding factor in quality is how pure the final ethanol/water mixture is. As for what the other things are that typically make it into cheap vodkas, I have no clue and would be happy to hear from others. But I wouldn't be surprised if it was a fairly complex mix of different chemicals.