Something contemptible is worthy of scorn, like the contemptible jerk who's mean to your sister; but contemptuous is full of it, like the contemptuous look you give that guy as he speeds away in his gas guzzler.

Both words turn contempt into an adjective. Contempt is a noun that describes the feeling that someone is beneath you or the state of being despised. (There's a legal meaning of contempt, but that's a question for another day!)

The -ible ending means "able to." Contemptible, then, means "able to be scorned, worthy of contempt." It refers to the person or thing the contempt is aimed at:

The contemptible crime of stealing a charity box resulted in a man being remanded in custody at Lerwick Sheriff Court today. (Shetland Times)

If familiarity truly breeds contempt, then money is the most contemptible thing in our lives. (Hamptons.com)

The educated classes are being swept along in the contemptible struggle for wealth. (Friedrich Nietzsche)

The –ous ending means "full of," so contemptuous means "full of contempt, showing scorn." It refers to the person or thing showing the scorn, like that contemptuous glare you gave that mean kid who dumped your sister, or in these examples:

Mr. Beck is, of course, free to be as contemptuous towards the president as he wants to be, but officers in the United States military are not. (Huffington Post)

Both men are now dismissive, even contemptuous, of each other's competence. (New York Times)

The reporter, gathering his wits, gave a contemptuous laugh. (Herman Landon)

Remember, then, if something is able to be scorned, it is contemptible. If someone or something is full of contempt, it is contemptuous. That contemptible thug made you feel contemptuous.