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.