Brightness refers to the percent reflectance of blue light, as measured at a wavelength at, or about, 457nm. The choice of that wavelength is based on the sensitivity of the human eye to blue and yellow light. Brightness was originally a test in paper manufacturing to measure the effectiveness of the bleaching process in removing yellowness from pulp. When paper is bleached, the spectral reflectance curve increases the most in the blue and violet range, at about the 457nm point. This has also made the measurement of brightness well suited for measuring the aging of paper because paper yellows with age. Most white papers are in the 60 to 90% brightness range.
Whiteness, on the other hand, refers to the extent to which paper reflects equally the light of all wavelengths throughout the visible spectrum. A truly white sheet of paper will not absorb one wavelength of light energy more than another. For example, if a sheet of paper is placed under a full spectrum light, most of that light will be reflected back equally and the paper will appear white. However, if some of the wavelengths of light energy are absorbed, the color of the paper will shift to the light which was not absorbed, but was instead reflected back to the viewer. That is why a red sheet of paper appears red in white light because it absorbs all the other colors and reflects only the red.
In the draft of the new ISO 12647-3 there is a value of 58% for paper brightness specified. The paper whiteness is "To be defined by the paper suppliers".