For the most part, modern Vector Signal Analyzers (including RTSAs) can be considered a superset of a spectrum analyzer functionality and vector signal analyzer functionality. Most modern VSAs can do all (or nearly all) of the measurements that a conventional swept spectrum analyzer can do. In fact, many manufacturers VSA/RTSA units are built inside of their swept analyzers - meaning that conventional spectrum analysis is handled by the conventional swept analyzer hardware, and the signal analysis is handled by a separate hardware path that is switched inline. There are common blocks, like the preselector for example. However, this also means that many of these analyzers are "mode driven", meaning that you are either in Spectrum Analyzer mode, *or* in Vector Signal/IQ Analyzer mode, *or* Realtime SA mode, etc. Also, for wideband signal analysis, the YIG preselector must often be bypassed...
Other manufacturers have built RSAs as a VSA from the ground up, without the conventional swept hardware. Even though this is the case, swept spectrum measurements can also be made, they're just done a little differently - basically by acquiring data within the realtime BW and computing the spectrum (in segments for wide spans) using FFT or Chirp-Z transforms. These analyzers don't give you a "sweep time" control because they're not sweeping like a conventional swept analyzer. However, the sweep times for wide spans with narrow RBW settings can be dramatically faster than a conventional swept analyzer. Also, these analyzers are generally not mode driven - meaning that you can view spectrum results simultaneously with time-domain vector measurements, modulation domain measurements, etc. - all time correlated.