Neither insidious nor invidious are happy words: insidious describes something that lies in wait to get you, and invidious is something offensive or defamatory. Cancer can be insidious, lurking in your body without your knowing it. Invidious doesn't hide; it's hateful right away.
Insidious didn't fall too far from the tree – it comes directly from the Latin word insidiosus meaning "deceitful, cunning, artful," from insidiae "plot, snare, ambush." Something insidious can even be attractive while doing harm, like an insidious plot to befriend your crush's girlfriend, so you can break them up. But often it's not attractive, just sneaky:
Storms like Agnes and Irene are insidious, often striking slowly over time in ways that can be unpredictable and far more damaging than anticipated. (Salon)
An insidious new email virus infiltrated high-profile US companies Thursday. (Toronto Star)
Rather, it is the insidious silence and insensitivity that surrounds so many of the most excruciating diseases of the mind that so often trigger suicide. (CNN)
Invidious comes from the Latin for ill will or envy. Some bouncers probably love the invidious task of not letting good-looking people into their clubs. It's often paired with segregation, but other things can be invidious as well:
Arnold is in an invidious position, and has tried to create a very different type of museum on the proverbial shoestring. (New York Times)
"After an old-fashioned, all-round team performance ... it might seem invidious to single out one player," admits the paper before singling out one player. (Guardian)
The cheap shots against the Democrats and Obama at the beginning were unnecessary and invidious. (Washington Post)
Joining the cheerleading squad so you can poison the football team is insidious. Yelling, "Teams like yours always lose!" at the game is invidious.