Before describing the difference, it is important to note that the human visual range is typically from 380nm to 780nm. Wavelengths within this range are not all treated equal, and wavelengths outside the range do very little in stimulating the human eye. Our sensitivity peaks at 555nm (Pure Green) and drops off at either side. The "photopic curve" defines the relationship of our eyes' sensitivity to wavelength and is illustrated on right.
Photometry is the science of measuring light within the visible portion of the electromagnetic spectrum in units weighted in accordance with the sensitivity of the human visual system. Photometry deals with perceptual issues. For example, if a surface radiates a given amount of energy, then how bright does that surface appear to an average viewer? By standardizing the luminous efficiency of the human visual system, the subjective nature of photometric measurement may be eliminated. This was done in 1924 by the Commission Internationale d'Eclairages or CIE, by performing empirical tests with over one hundred observers. The CIE material was proposed in 1924 and adopted as an international standard in 1931.
Radiometry is the science of measuring light from any part of the electromagnetic spectrum. In general, the term usually applies to the measurement using optical instruments of light in the visible, infrared and ultraviolet wavelength regions.