Yes, there are quite a few esters and acids that will hitch a ride with some of the alcohol and water molecules that make it up the column. This certainly depends on the type of still (eg plate vs alambic and continuous vs pot), but there are a lot of flavors transfered. The distiller's job is to choose which ones (eg at least remove the heads and tails).
One easy example is apple brandy which has a very distinct nose. It's not until you multiple distill and charcoal filter that vodka loses its flavor (you can actually detect potato, if you care to).
I should also mention that low pH (acidic) environment is important to the proportion of many flavors being transmitted. That means that distillers (looking for flavor) will often prefer lower alcohol but fully fermented brew, which usually means starting with lower sugar (less ripe fruit or less malted grains). Corn usually starts high sugar so I'm not sure about them.
The types of grains certainly do; rye, bourbon (corn), and un-peated scotch (barley) taste different. I don't know about the types of corn, though; the types of barley don't seem to make a big difference for scotch.
I live near a small distillery that does tastings and tours. The mash composition is a huge component of the flavor, and it's a lot of fun to taste the different mashes pre-distillation, along with the final product. The percentage of a corn to rye makes a really big difference, but they even pay attention to the source of the grain they use.
Also the new mash smell is the greatest thing in the world.
four roses produces single barrel bourbons - typically through their barrel select program, where you can taste a couple of different mash bills, but more important to the question: yeasts. you can taste the different yeast strains changing the end flavor. everything makes a bit of difference, if you're you're not trying to make neutral grain spirit.
I was under the impression that most flavor comes from the organic material present in aging barrels.