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They're all used for different purposes so you really can't talk about advantages or disadvantages. If you need to take a particular measurement generally only one instrument will do it well. Here is a brief overview of the different instruments.
An oscilloscope is the most basic and it only measures a signal's amplitude over time. Then comes the spectrum analyzer which measures a signals frequency spectrum, a vector signal analyzer adds demodulation to determine the signal phase. The network analyzer adds a signal source and measures the phase and amplitude response through a component, module or system.
This is the short answer but I'm Wikipedia has better explanations than mine.
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