Luminance is the luminous intensity, projected on a given area and direction. Luminance is an objectively measurable attribute. The unit is 'Candela per Square Meter' (cd/m2). So, different monitors can be adjusted to the same luminous intensity by measuring the luminance in cd/m2. To learn more about Luminance, visit http://www.crompton.com/wa3dsp/light/lumin.html.
Brightness is a subjective attribute of light. The monitor can be adjusted to a level of light between very dim and very bright. Brightness is perceived and cannot be measured objectively (but scaled, e.g. in %).
So, what is a colorimeter or spectrophotometer doing when it adjusts the Brightness? It is evaluating the black point, which is the starting point for the Brightness curve. Two black patches, which are very close together, will be measured. The best level for the Brightness has been found, as soon as the measurement device can measure a slight difference between the two black patches. The Brightness level would be too low, if no difference is measurable, and it would be too high, if the difference would be too big.