The classic Voronoi Diagram use-case is phone-booths on a campus. Let's say you need to use the phone and you want to know which phone is closest to your current location. A map that shows you this in the simplest way is a Voronoi diagram where the points are phone-booths and the polygons are the area around each phone where that phone is the closest one. Find out which region you are in and you know which phone to head for, no measuring required.
Generally speaking I think Voronoi diagrams are useful any time you have a collection of points and you want to visualize them as geometric areas instead of clumps of points.
My use-case isn't so different from the campus phone maps, but I don't want to get into specifics. Our competitors are everywhere!