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Basically they are the same thing, and represents a phenomenon that appears as an input signal on the screen of the analyzer even though there is no real input signal (appears internally, when the signal and its harmonic components are mixed).
The difference is that residual response is an absolute value defined in a particular bandwidth, when spurious response is a relative to carrier value which includes all the mixing products that appears.
For a user of the analyzer more important are spurious response values.
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