In my experience, visual borders have no meaning.Are all of your white dots exactly 255,255,255. I ask because GIMP default is pressure sensitive clicking, so a light click only gets you part of the way to 255,255,255.
You don't need visual borders, but doing a map without them if the provinces are irregularly shaped will make navgating the map a nightmare. I should know, the first Dom2 Faerun map was like that.