Photometers record the flux of light in a given range of wavelengths from a given part of the sky. Filters in the light path allow selection of the wavelengths studied. Photometes are especially useful for studying objects that vary in brightness, especially in short periods of time. Spectrometers break light into component wavelengths. Usually use a diffraction grating, which bounces light off of a finely ruled mirror.
Different wavelengths diffracting off the lines are bent by different amouns, producing a spectrum.
Spectrometers are best for doing detailed analysis of an object. Imagers produce a 2-dimensional image of the sky. These are best for determining shapes of objects and finding the relationships between different objects. Before the late-19th century, all astronomy was done by drawing what the astronomer saw. Through most of the 20th century, astronomers took photographs of spectra and images.