Trust me, I don't always agree with Mr. Wienke but in this case, I do, pretty much 100%.
A "Vector Network Analyzer" is as opposed to a "Scalar Network Analyzer"; the former looks at BOTH the magnitude AND the phase of the reflected signal versus that of the incident signal and bases its results on the difference while the latter looks at ONLY the magnitude of the reflected signal versus that of the incident signal and, again, bases its results on the difference.
In both of those devices the results can be computed and shown in many ways including VSWR (a VSWR meter is a very specific and simplified form of a Scalar Network Analyzer) but a VNA is MUCH more telling and flexible. Knowing the phase of your reflected signal relative to the incident given a normalized calibrated origin point (usually 50 ohms resistive for most RF testing) can tell you exactly in which "direction" (a loose analogy) to "tune" for in terms of capacitive or inductive reactance to add to in a matching circuit or for lengthening or shortening an antenna element.
As for "feedline and connectors", yes, well you definitely want to check those but it's always best to do those as separate tests. Feedlines can be measured in a two port analysis using S12 and S21 measurements and connectors can be part of that feedline, usually. For the test lead to the antenna (or whatever single port "Device Under Test" or "DUT") you use your calibrated test cable, preferably, and run your open, short, and load calibrations with the cable in place to effectively make the test lead "invisible" to the final results for the single port DUT.
Anyway, went beyond what I think is needed here, but the idea is a VNA is absolutely a "full blown network analyzer" and, while a "return loss bridge" may be required for some older forms and for devices not actually intrinsically network analyzers (such as using a spectrum analyzer and a tracking generator) modern full featured VNA's already include that functionality built in. It doesn't need to be mentioned in the documentation because you now only need to focus on the actual one and two port S parameter analysis, S11, S22, S12, and S21.
-Mike