A Vector Network Analyzer (VNA) generically measures the scattering parameters between ports in a multi-port network. These could mathematically be converted to Impedance Parameters, but this will necessarily include transfer impedances as well as the input port impedances.
An Impedance Analyzer on the other hand emphasizes the accurate measurement of complex impedance between a pair of terminals. I don't believe transfer impedance between ports is generally included in that measurement.
When a network comprises only one port, I suppose there is not much difference in principle, but I think usually the methodology is different and therefore the accuracy will be affected depending on what you are trying to do. If, for example, you are trying to measure a very high impedance, say 10 kOhms, the VNA will give an accurate reflection coefficient very close to 1, but the impedance you calculate from this could be a off by a large fraction. The Impedance Analyzer on the other hand, depending on the frequency, may use a current source and sensitive voltmeter to detect the impedance and give an accurate result. Such Impedance Analyzers I believe are much more limited in frequency, however.
you can make more measurements with a VNA . You can use the S nn port measurements and you can derive the impedance and very more constants and values. With a impedance measurement device you can only measure the impedance and not more.
VNA can measure complex reflection and transmission coefficients. From that you can derive impedance an many other parameters. VNA is usefull in microwave networks, however, depending on your model and frequency range of your VNA, it will be uselfull at low frequencies, such as kHz or lower. VNA is a high accuracy measurement device. Impedance analyzer uses the same principle for measuring impedance, but this is a low frequency device. It can performs as well as a VNA at low frequencies. Which one to use? Depends on your measurement requierements.
Basically, both measures inter related parameters. However, impedance measurement is preferred at low frequencies and no VNA available at low frequency (Also not required, in reality - its economic trade off). VNA is used at high frequency as requirement of parameters is not only impedance but complex parameters. Both has their importance at different frequency range
VNA's measure s-parameters and impedance analyzers measure impedance. But S-parameters can be converted to impedance, so what's the difference?
A major difference between VNA's and Impedance analyzers is that the VNA uses a reference impedance, normally 50 Ohms, that can be easily matched to a transmission line. That allows you to use cables and PCB fixtures that have matched lines, so that you don't have to worry about parasitic inductance and capacitance in your setup. The higher the frequency, the more this matters.
As others have said, you can convert S-parameters to impedances, but the reference resistance is always 50 Ohms. This means that impedances that are far from 50 Ohms will be measured with large errors.
An impedance analyzer can modify its reference resistance based on the required measurement. This improves the accuracy of the measurement, but it means that the cables and fixtures behave as parasitic elements.
Thus the VNA is best for high frequencies and the impedance analyzer is best for low frequencies. Typically impedance analyzers are used from DC to 10s of MHz, and VNA's are used from MHz to 10s of GHz.
As others have noted, VNA's can measure multiports. But if you look at the app notes for impedance analyzers, they can too. A transformer is a common example.