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.
I thought this article looked credible and gave a thorough explanation:
Part of what makes this confusing for a lot of people is that all spectrophotometers incorporate a spectrometer. It’s also true that other analytical instruments use spectrometers. The spectrometer is the part of the spectrophotometer that is most responsible for measuring things. The spectrophotometer is a complete system that includes a light source along with a means to collect the light that has interacted with the things being tested, as well as a spectrometer for measurements.
I think the article explains both a spectrometer and a spectrophotometer in detail which may give more insight than simply the comparison.
The article addresses why it can be a little confusing (as well as a detailed answer), but in short, a spectrophotometer has a spectrometer in it, but the spectrophotometer is an entire system.
I believe this image/vector belows shows an entire spectrophotometer, where as the gauge at the end I believe is only a spectrometer.
Furthermore, I thought providing a concrete example in industry may be useful. In laser safety, lenses are tested with a spectrophotometer. The spectrophotometer will test the amount of light being absorbed against the amount of light being let through at each wavelength in question. The amount being absorbed is a measurement known as Optical Density. See the diagram below for an brief overview on how a spectrophotometer works. Please note, the spectrophotometer depicted below is showing a spectrophotometer for visible light; different spectrophotometers test for different wavelength spans (UV and IR).