Well with a common cold the soreness is cause by your immune system overreacting to the cold virus. Excess coughing helps spread the virus, and so viruses that tend to trick you into coughing a lot tend to spread more and be more successful as viruses. The soreness is a byproduct of inflammation and damage caused by the coughing.
Flu is a bit different because cells tend to self destruct as a way to prevent the virus from using them to replicate. So you have pretty much direct cell damage in addition to the excess coughing causing sore throats.
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