I've noticed that YouTube's autoplayexperiment shows up more and more often, which usually means that it's ready for release. One of the changes in the latest iteration is that the player has a "next" button, which lets you play the video from the "up next" section. There's no "previous" button, but you can use the browser's "back" button.
"When autoplay is enabled, a suggested video will automatically play next," explains YouTube. If you don't like this feature, you can disable it from the right sidebar or from the player's controls.