Inappropriate Sims will remain unaffected by a rude house guest. They can also make fun of the nearest person standing nearby to the Sim they're having a conversation with. (That's why trait is perfect for mean-spirited and evil Sims!)
EDIT: Inappropriate Sims can also make inappropriate forum posts, when using the computer. It's similar to Trolling on the forums, which exclusively for Grumpy, Rebellious, Evil and Mean-Spirited Sims.
The trait description claims that Inappropriate Sims will not dress properly yet I see no evidence in the game. Perhaps EA made a mistake, and meant to write the "never think to dress properly" part of the description for the Insane trait.
http://sims.wikia.com/wiki/Inappropriate