Harmonics is multiple of fundamental frequency. Spurious is not related to fundamental frequency but it is observed at the output spectrum. Harmonics are generated due to non-linearity of the device. Harmonics can generate spurious. As an example harmonics can produce resonance with mechanical box internal structure and can generate spurious.
Harmonics generally generated from TCXO & frequency multipliers. If amplifier is non-linear, this will also generate harmonics. But harmonics very near to carrier generally generates interference to modulating signals. We design final power amplifier as a class AB amplifier to get more output power and better efficiency and also to minimize output power variation with temperature. Since it is non linear amplifier it generates harmonics but it will be very far from carrier and can be filtered easily.
See the link for detail :- http://harshit.org/spurious and harmonics.pdf
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