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Spectrometry is virtually a synonym for spectroscopy or spectrography. For more information, see spectroscopy.
The term 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, in advanced instruments, 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.