Nowadays there isn't that much difference.
A signal analyzer has the ability to demodulate RF signals....either with internal DSP processing, or at least by digitizing the I and Q data and providing that for external processing on a laptop. They often contain basic spectrum analyzer capabilities too, like displaying spectrum, spurious signals, power level, but are not as well qualified to do so as a real spectrum analyzer. Signal analyzer are often cheaper.
A "spectrum analyser" is optimized to show accurate power vs frequency data. Some of them can also demodulate or digitize the signals...but maybe only 30% of them have this capability.