The main advantage of a true spectrum analyzer is probably dynamic range.
Using digital sampling, you have an inherent noise floor due to discretization noise. This floor is somewhere around 1/2n of your peak-to-peak signal amplitude, where n is the bit size of your discretization. For 8-bit sampling this about -50 dB below your signal level.
Using rf techniques you can easily (but not cheaply) find instruments capable of 150 dB or more of dynamic range.
Also down-mixing may allow more cost-effectively testing high frequency (GHz and up) signals, which would require excessive sampling bandwidth to test with an FFT-based scheme.