Spectrum analyzer types are distinguished by the methods used to obtain the spectrum of a signal. There are swept-tuned and fast Fourier transform (FFT) based spectrum analyzers:
*A swept-tuned analyzer uses a superheterodyne receiver to down-convert a portion of the input signal spectrum to the center frequency of a narrow band-pass filter, whose instantaneous output power is recorded or displayed as a function of time.  By sweeping the receiver's center-frequency (using a voltage-controlled oscillator) through a range of frequencies, the output is also a function of frequency.  But while the sweep centers on any particular frequency, it may be missing short-duration events at other frequencies.
*An FFT analyzer computes a time-sequence of periodograms.  FFT refers to a particular mathematical algorithm used in the process.  This is commonly used in conjunction with a receiver and analog-to-digital converter.  As above, the receiver reduces the center-frequency of a portion of the input signal spectrum, but the portion is not swept.  The purpose of the receiver is to reduce the sampling rate that the analyzer must contend with.  With a sufficiently low sample-rate, FFT analyzers can process all the samples (100% duty-cycle), and are therefore able to avoid missing short-duration events.