The photometer just measures the intensity of light. Think of a solar cell hooked up to a voltmeter. The higher the intensity of the light, the higher the reading on the voltmeter ( a simplification, but you get the idea).
A spectrophotometer has a more sophisticated system of separating the light into the individual wavelengths and measuring the intensity at each individual wavelength. Typically, a spectrophotometer has a monochromator (for separating the individual wavelengths of light) and is a scanning instrument. As an example, it will scan from something like 400 nm to 200 nm at 100 nm per minute.
A spectrometer is a scaled down spectrophotometer. It can use diffraction gratings to separate the individual wavelengths. Commonly, this type of instrument is used in fourier transform (FT) instruments.
These instruments can be for many ranges of light, including IR, UV, VIS, etc.
Theoretically, a spectrophotometer is the best. Don't always count on this. Many of the spectrometers offer faster speed and similar resolution to spectrophotometers. They are often cheaper, using computer power instead of optics to separate the wavelengths of light.