They are not quite synonyms. "spurious" is an adjective which relates to theories and arguments, and means unconvincing, weak, untrustworthy e.g. His argument that the sun goes round the earth was spurious. "sham" is a noun and an adjective which means counterfeit, fake and normally applies to events or situations which have been planned in order to deceive e.g. sham marriages

"Sham" implies deception. It implies a false front or a pretense. "I thought he was on my side, but it was all a sham. He was just pretending to be on my side. He was really working for my enemies." "Spurious" means invalid or misleading. "The car alarm was set off by a spurious signal from nearby lightning bold."

The meaning is very similar, but "sham" is a noun, while "spurious" is an adjective, so they cannot be used interchangeably.