Why do distilled alcoholic beverages taste different from each other?
Two different aspects come into play here. For once, most of the whiskey aroma stems from aging the fresh destillate in oak barrels previously used to store wines, e.g. sherry. Colour an aroma thus, at least in part, stem from a  slow extraction process using the fresh destillate as a solvent.

But thinking in other beverages, particular the clear fruit spirits, there's more in it. The key to a beverage that doesn't smell and taste just like ethanol but has a cherry aroma is the sloppyness of the destillation :) In the lab, a destillation or rectification is performed to isolate a particular compound from a mixture at a sharp boiling point. 

In the production of alcoholic beverages, the destillation typically aims to obtain three fractions over different boiling point ranges. 


At lower boiling points, methanol, acetone and ethylacetate are removed.
The upper range mostly furnishes long chain alcohols.


These two fractions aren't meant to be digested ;)

The central fraction, however, will contains all the fragrances, ethanol and water.