For something to be specious it has to appear to be correct or true, but in reality is false or incorrect. There is a slight connotation of it being related to appearances, that the specious item has an allure that is untrue.
Its derivatives include speciously and speciousness.
Spurious, on the other hand, is something that is founded on illogical reasoning or false facts. It can be something that is similar in appearance to something else, without being the genuine item (e.g., drugs). This usage is found commonly outside North America, specifically India and the Middle East.
Another definition of spurious includes someone who is born illegitimately. Medically, spurious refers to symptoms which seem like something else, but which have no cause.
Its derivatives include spuriously and spuriousness.