A person who feels contempt toward something else is contemptuous toward that thing. Something deserving of contempt is contemptible. If you regard something as inferior, base, or worthy of scorn, you are contemptuous toward that contemptible thing.
Contemptuous is synonymous with disrespectful, arrogant, and condescending, while contemptible is synonymous with despicable and worthless. Contemptible is usually overtly negative, while contemptuous is often more subtly negative, implying that the contemptuous person is unnecessarily arrogant.
The far left is enraged and talking impeachment; the right is smug and contemptuous, even when they agree. [The Atlantic]
David Cameron’s reaction was contemptible: he stuck his fingers in his ears and refused to recognise the fury his policies have aroused. [Mirror]
Daisy might have been contemptuous of convention, but she was not prepared for the decadence of the British upper classes. [Wall Street Journal]
Schiller and the guerillas trade riffs about how contemptible members of the Tea Party and the Republican Party are. [Slate]
Diplomats normally contemptuous of the views of the Arab League suddenly treat its call for a no-fly zone as evidence that the Arab world favours intervention. [New Zealand Herald]
This venomous, contemptible cabal has single-handedly moved the needle on revulsion. [The Augusta Chronicle]