What actually happens to reduce the perceived color in a 'white rainbow" or "fog-bow"? The image below is included in the BBC article Amazing white rainbow snapped over Scottish moor. Apparently this phenomenon is sometimes called a "fog-bow" and is characterized by the washing out of perceived color relative to a conventional rainbow.

Is this related to the geometry (fog being far closer to the viewer) or properties of the droplets themselves? If so, which properties and why? 

Higher resolution in flickr
The reason why fog bows lack colour, compared to rainbows, is due to the size of the drops of water.

Fog is composed of very small drops of water - less than 0.05 mm diameter. Because of this the wavelength of light is critical, with diffraction smearing out colours that the larger drops of water in a rainbow would make.