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.
There is much talk, on the one hand, of “confronting Ukip head-on”: attacking its specious argument about the damaging effects of “mass migration” (a once universally accepted term that has itself now come under attack) on the lives of indigenous communities. [The Telegraph]
The Lexicon Valley podcast brings in experts who present the interesting but probably specious idea that starting a sentence with “so” has its origins among Bay-area techies, but my experience would suggest it’s just a regional way of speaking that spread, as colloquialisms tend to do. [KMUW]
Football’s ultimate puppet master staged and practiced the speciously spontaneous scene that celebrated the historic touchdown, as fans holding giant cardboard numbers 5, 0 and 9 danced in the stands. [Denver Post]
It then dabbled in what can only be called bureaucratic speciousness, claiming the group had asked to see a “traffic study”, whereas it says it had only conducted a “traffic assessment”. [South China Morning Post]
It may be noted that four days ago, excise officials arrested four persons in a bar and liquor shop at Srikakulam while they were selling spurious liquor by filling it into water bottles. [Deccan Chronicle]
Secretary Health Jawad Rafique Malik directed that a vigorous crackdown be launched against the elements involved in the nefarious trade of spurious and substandard drugs and the steps taken in this regard should be visible on ground. [Pakistan Observer]