title_body: How much of a spirit's flavour comes from the original fermentation?
upvoted_answer: The type of still used is significant. Vodka is distilled using a fractioning column or reflux still as described here:
  
  https://en.wikipedia.org/wiki/Fractionating_column
  
  With a fractioning column, the answer is: none of the flavour comes from the original fermentation. There is no flavour (sort of....)
  
  Here's what a commercial one looks like:
  
  
  
  Or a home use version: 
  
  
  
  I'm not very au fait with exactly how a reflux column works, but the end result is a pure flavourless liquid. Roughly the vapours travel up through a long column which contain elements that prevent all but the purest alcohol vapours to pass through.
  
  In the case of Whisky, Rum, Brandy, then yes they do get their flavours from the original fermentation. They are distilled using a pot still (by law in the case of Scotch and Irish whisk(e)ys). This is a much older type of still, less efficient than a column (lower ABV and less yield); key thing is it retains flavor:
  
  
  
  Or a home use version:
  
  
  
  The nature of a pot still is that it is less efficient than a column, but it allows much more flavor to emerge. 
  
  It is possible to make vodka on a pot still, but great care has to be made to ensure the input alcoholic liquid (the "wash") is as clean and neutral as possible, no off flavours from the yeast for example. No commercial distiller would attempt this but some home distillers make do with what they have.
  
  Note that even whisky, which is distilled from dark brown malt "beer", will come out of the still as clear as vodka. It derives its final color from the barrel ageing process.
  
  Re the rums, I think techniques vary but I think usually white rum is similar to dark rum but it is carbon filtered after the oak aging process is complete, which removes the colour and also presumably some flavour.