Spectroscopy: you pass the light through (or reflect from) a dispersive element (a prism or diffraction grating) and then you record the dispersed light. You have a record of the intensity of the light as a function of wavelength. Advantage: potentially you can record the light of one or more objects over a very wide wavelength range and have excellent fine discrimination between different wavelengths and can look for individual spectral features. Disadvantage: You spread the light from the source thinly over the detector and it can be difficult to obtain spectra of many sources in one go (maybe 10s to 100s if you use multi-slit of multi-fibre spectrographs).
Photometry: you record images of your sources where the light is allowed to pass through coloured filters. The only wavelength information you have is the intensity of the light admitted through the filter (and also modified by the detector response). Advantages: The light from a source over your filter band is concentrated onto a spot on your detector - giving better signal-to-noise ratios.
You also get data for as many sources are in the image - potentially thousands.
Diasadvantage: The effective wavelelength resolution is only as good as how narrow the filter bandpasses are. You lose the ability to see individual spectral features.
These days the detectors used for both techniques is usually a CCD camera. So what differs is what is in the instrument prior to the CCD. For spectroscopy it is a dispersive element, for photometry a coloured filter.
Both techniques can be used in extragalactic astronomy to estimate redshifts. Taking spectra is far more accurate, but a rule of thumb is that photometric techniques, though comparatively uncertain, can be applied to objects a couple of magnitudes (a factor of 5) fainter. This rule of thumb applies to objects with a broad spectrum and strong continuum - most stars and galaxies. For certain types of sources with very strong emission lines, where the power of the source is concentrated into wavelength ranges much narrower than photometric filters, it can be the case that spectroscopy is more sensitive even in the same observing time (e.g. some types of AGN and quasars).
To get a redshift from a spectrum is a fairly obvious procedure. You try to match the positions of known spectral features and measure the wavelength shift from the restframe. Estimating a redshift from photometry needs images taken through several different filters and the "spectral energy distributions" (the crude intensity vs wavelength relation defined by the few photometric brightness measurements) are matched with those predicted from a library of model galaxies redshifted by different amounts.