In astronomy, a spectrometer takes the light from an object, passes it through a slit, which then defines the input to a dispersive element. As a result a (generally unknown) amount of light is lost outside the slit, so it becomes difficult to estimate the absolute flux as a function of wavelength.
In a spectrophotometer you do not use an entrance slit, so all the light is collected and it becomes easier to estimate the absolute flux spectrum. The penalty is that the resolution is worse and is dependent on the size of the input image (or "seeing " in the case of stars).
A spectrometer measures emission spectrum, as the light is focused on its entrance slit, dispersed and registered at different wavelengths.
A spectrophotometer measures absorption spectrum of a sample placed inside. The light from a built-in broadband light source is dispersed, sent through a sample and registered at different wavelengths.
I really like this description straight from the optical spectrometer wikipedia page:
Spectrometer is a term that is applied to instruments that operate over a very wide range of wavelengths, from gamma rays and X-rays into the far infrared. If the instrument is designed to measure the spectrum in absolute units rather than relative units, then it is typically called a spectrophotometer. The majority of spectrophotometers are used in spectral regions near the visible spectrum.