Wadi Rum is around 344 km south of the Dead Sea and takes about 4 hours by car journey. I recommend if you stay by the Dead Sea, you allocate an entire day (on your 2nd day) to visit Wadi Rum (and Petra on the way back). You should start early in the morning, preferably by sunrise.
Referred to as The Valley of the Moon, Wadi Rum is a valley that has been cut into sandstone and granite rock and has been inhabited by several human cultures since prehistoric times. They have left their mark in the forms of temples, graffiti and rock paintings. Here is where they filmed the movie “Lawrence of Arabia” with Peter O’Toole, depicting the true story of the British officer T.E. Lawrence who passed through this valley many times during the Arab Revolt in 1917-1918.
This area is now one of Jordan’s important tourist destination, including for trekkers, climbers and camel and horse safaris. Some tour operators offer overnight luxury camping treats as well. The views that you get in many areas surrounding Wadi Rum are spectacular.