Bart, the Cube is essentially a tristimulus colorimeter. As such the photodiodes aren't associated with 'filter bands' on a spectrum per se. Rather, the tristimulus values are derived from a combination of 12 different outputs, each of which can be interpreted to represent a section of the spectrum. Its just that the final output isn't derived from a spectrum. Because the effective light source is identical under each sample, colors with different spectral distributions but which appear the same (ie. metamers) will result in the same output.
hjulenissen, you're right. The use of the spherical enclosure is to ensure the same reading regardless of surface sheen. We found many existing colorimeters don't function well on different sheen levels of the same color, and that this would be an issue when matching, say, the color of a polished tile. Some of the devices made for paint matching also needed to be told whether the surface was shiny or dull (which in itself is quite subjective). For print calibration, this is perhaps less of an issue given the choice of paper is arbitrary, so a matte piece of paper can be used for calibration.