I've used the built in FFT filter in Adobe Audition to get some vocals sounding like they're coming from a police radio, and while the black box worked, I'm just curious as to what the FFT filter is actually for, and what differentiates it from an equalizer? What cases calls for one verses the other?
An equalizer is a combination of filters with different shapes that you adjust in some constrained way.
The FFT filter is based on the Fast Fourier Transform, which is a different way of manipulating signals. It can be viewed as having a direct control of the amplitudes of a selected number of bands (e.g. 2048 bands) in the frequency domain. In other words, it's a lot more precise type of equalization. It can be used to get those vocoder-like filtering effects as well as for noise reduction or very precise equalization. As it involves heavier calculation, it has a considerably larger latency and can produce unintentional processing artifacts in real-time use.
Use an equalizer as you'd use it normally.
Use FFT equalization for getting more precision that an EQ gives, for correcting very small frequency areas (e.g. supressing noisy parts or very small peaks), for performing equalization of some custom shape that your EQ cannot do or creating a fine-tuned, static filtered sound.
The algorithm is different. Typical eq works in a simmilar way that analog circuits does, but in a digital domain. There are FIR (finitive impulse response) and IIR (infinitive impulse response) filters, not going to much in technical details.
FFT filter makes fourier transforms of the sound to frequency domain, where you can easily acces distinctive frequency bins and the amplitude of each frequency can easily be changed, and after that - inverse fourier transform is made to go back to time domain.
An "FFT filter" is not an established term. The main purpose of FFT is to speed up convolution with an ongoing signal (cf "overlap-add" and "overlap-shift" algorithms), so I expect the "FFT filter" to just be an implementation of a long-response FIR filter that is rather efficient at the price of considerable time lag. For offline processing, the time lag is not a problem.
A long-response FIR filter can be used as an equalizer of course since an equalizer is a linear system. However, equalizers tend to rather use IIR filters (preferably minimal-phase) or quadrature mirror filters for their operation, either of which lead to smaller systematic delays than an FIR filter, and they also map nicer to logarithmic frequency scaling which is customary with equalizers used for musical purposes.