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We sample a signal over time (e.g. voltage fluctuations of a resistor). Then do a FFT-based Power Spectral Density in a PC.

What advantage would a physical spectrum analyzer provide over doing the spectral analysis numerically in a PC? (I am NOT interested in a real-time analysis, I only do post-analysis)

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None, if the front ends and samplers and algorithms are equally good. A number of recent spectrum analyzers are PC's, with custom front ends and an enclosure and display/controls suited to bench use. Typically though, a packaged spectrum analyzer gives you a front end, converter, sampler, and algorithms which it would be quite a project to duplicate from individual components. –  Chris Stratton Aug 5 at 20:27

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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.

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I'm under the impression that pretty much every modern SA is ultimately sampled, so it's really a question of how good a sampler, and how much the instrument shields the sampler behind a narrow filter it uses mixers to move the signal past, vs how much it uses digital processing to pull a given frequency bin out of the sampler bandwidth, with the same instrument behaving differently depending on its settings. Some of these tradeoffs are explicit in the user interface, others seem derived from more goal-oriented user controls. –  Chris Stratton Aug 5 at 20:48
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@ChrisStratton, I'm not surprised if that's true. But there's still a difference between trying to sample a 50 GHz signal with 16-bit precision directly, vs using some rf stages to downmix that to 100 MHz or so where high bit-width sampling is more reasonable. –  The Photon Aug 5 at 21:23
    
Agilent/Keysight has service manuals for some of their instruments online, including block diagrams for the internals. For example, in the N9030A analyzers, it looks like they mix everthing down to 22.5 MHz before digitizing at 16-bits x 100 MSa/s. –  The Photon Aug 5 at 21:37
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@ChrisStratton: I believe most modern SA's do indeed sample at some point in their block diagram, but the "distance" of the ADC from the front end varies from analyzer to analyzer. The Photon is correct that some Keysight products put the ADC after all of the down converting. –  Ryan Aug 6 at 1:02
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@Ryan - yes and no. The linked example samples an IF, which will then be digitally downconverted to 0 for IQ processing. Nobody is going to build a lab-grade SA with a bare ADC, and nobody is going to do it all in analog any more. How much is done by scanning the LO and selecting filters before the ADC, and how much by digital processing after varies, usually including between various settings of the same analyzer. –  Chris Stratton Aug 6 at 1:04

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