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Spectrum Analyser gives a graphical display of the frequency content of the signal you are observing. It tells you the magnitude(s) and frequency(s) of the signal. So if you were to look at a simple sine wave with a frequency of say 1 MHz and amplitude of 1V RMS, you would see on the SA display a vertical line at 1MHz with a height of 1V RMS (usually vertical scale is in dBm or dBµV)
Vector Scalar or Network Analysers also displays data on its screen, but it gives you information about the characteristics of the circuit. So for instance if you had an amplifier and you wanted to measure its gain/loss characteristics with frequency (like a Bode plot), then you could use a scalar or network analyser to do this. The actual analyser provides the input signal to the Device Under Test (DUT in this example is an amplifier) and measures the output signal from the DUT. The output signal/input signal vs. frequency is the gain of the amplifier.
Other useful things the scalar or network analysers can do is give you a measure of DUT input/output reflection parameters (usually referred to as S-parameters) or even input/output impedances of DUTs as function of frequency. Very useful when designing RF amps etc. eg HP8753 is such a machine.
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