Why SWR is a ratio derived from forward and reflected power or from the complex impedance seen at the feedpoint, and all three instruments can get there. A directional wattmeter separates forward and reflected power, and SWR = (1 + sqrt(Pr/Pf)) / (1 - sqrt(Pr/Pf)). An antenna analyzer injects a low-level signal and reports SWR (and often R and X) versus frequency. A vector network analyzer measures the complex reflection coefficient S11, from which SWR = (1 + |S11|) / (1 - |S11|).
Watch out It is tempting to pick just the antenna analyzer because that is the hobbyist's usual SWR tool, but the other two instruments measure the same quantity, one from power ratios and one from reflection coefficient.
Anything that can see reflected power or reflection coefficient can give you SWR.