You can think of a spectrum analyzer as a receiver that constantly scans. You can also scan zero span and demodulate AM and FM signal on some models. A major con I can think of are, you really don't want to keep an antenna connected to SA at all times. If you get static coming in, it is so easy to blow the input section, and that is a majorly expensive proposition to repair it. Also, demodulation is limited to AM and FM. You can't demodulate SSB.
Pro, of course is speed and bandwidth of scan. You can scan an entire spectrum range in one sweep on SA, where as with SDR, it is typically limited to 50MHz or less at a time. With SDR, prices tend to be lower, and it is designed to be a receiver, so having antennas connected is not typically a problem.
Which is better totally depends on your purpose. There is not much justification to go the spectrum analyzer route if all you want is to receive a single signal at a time. If you need quantitative measurement, then SA would be more appropriate.
Hope this helps