Spectrometry is virtually a synonym for spectroscopy or spectrography. For more information, see spectroscopy.
It originated as a general term for the use of a spectrometer, that measures intensity as a function of wavelength, with spectroscope a more specific term for an instrument you "look through" (like a microscope or telescope) and spectrograph for one that produces graphical output. Now virtually all data is collected digitally, easily displayed or printed graphically, so the term chosen for a specific instrument is insignificant.
Mass spectrometry is an analogous study of the relative amounts of particles based on their mass, i.e., the use of a mass spectrometer. In this usage, the convention is to use the word spectrometry rather than the others.