What's the difference between analog and synthesized signal generator?

I think the main difference is the phase noise.
Synthesized will allow for a very low phase
noise source.

Analog signal generator is usually a free-running oscillator which frequency is controlled by a L/C network. Stability is not very good, needs long heat-up period. Phase noise can be good to very good. The venerable HP8640 had excellent phase noise far from carrier. This generator could even be frequency locked.

A synthesized generator usually involves PLL (Phase Locked Loops) and/or direct digital synthezier, DDS. The result is a high resolution high purity signal. Locked to a reference it can be as stable as needed and phase noise can be excellent. Drawback is the complexity vs. the analog signal gerator.

The title of your question mentions a spectrum analyzer, and the above are valid for sweeping local oscillator in a spectrum analyzer

To summarize the differencies: Performance, complexity and price :)