Insidious and invidious both describe the way someone does something bad. Insidious implies secrecy, while invidious is more open. An insidious action is sneaky and harmful. An invidious action is also negative, but it happens out in the open.
Insidious actions are covert. Those who carry them out hope to accomplish some act of evil. For example, lies are insidious. Secret meetings and intrigue are insidious activities. Insidious behavior typically isn’t bad on its surface. For example, making friends with a classmate is positive on the surface. It becomes insidious if the only reason for the friendship is to somehow use the other person to advance some agenda.
An action is invidious when it sows discord or strife between people. Invidious behavior can also be hurtful for no clear purpose. Invidious comparisons drive a wedge between people by comparing them in an unfavorable way. Invidious propaganda sows hatred and mistrust. Unlike insidious words and actions, invidious behavior happens in public view. Onlookers almost always recognize it as negative when it happens.
Both insidious acts and invidious behavior are always negative. Where the two terms differ is in their approach to doing harm. The open attack of invidious words and actions is different from the sneak attack of an insidious person. To prevent confusion, you can remember that insidious contains an S, as in sneaky, while invidious has a V as in vocal or voiced.