In addition to single frames shot using "focus and recompose" there may be times when you want to focus and then take multiple frames without refocusing as conditions in the viewfinder may change between each shot. By selecting Single AF and locking in the focus using the camera's back body focus button the camera will hold that focus as you shoot several frames. If you have Continuous (Servo) AF selected, holding the back button down may allow the Continuous focus to remain active while changing focus as the conditions in the viewfinder change, depending on how your back button for focus is set up.
If it is set to 'lock focus' (default for Nikon) it will function the same and lock focus in either C-AF or S-AF. If it is set to 'start focus' (default for Canon) it will not lock focus in C-AF/Servo AF mode but will start and lock focus in S-AF/One Shot mode. Some users choose to set the AF-lock/AF ON button to 'start focus' so the half shutter button press can be set to only start metering without engaging AF as well. With Canon cameras there is also the option to remap the 'Depth of Field Preview' button or the Super Telephoto lens based 'AF Stop' button to toggle back and forth between 'One Shot AF' and 'AI Servo AF' modes.
Imagine a situation where you are photographing a dance team doing their routine. They are lined up in three rows. You select an aperture appropriate to fit both the front and back row into the depth of field (DoF) and select a focus point for one of the dancers currently in the middle row. But as you are shooting the dancer you have focused on moves right or left or to the back or front row. If you don't want to change the overall framing of the shot and you have Continuous AF selected then you might need to move your focus point to another dancer now in the middle of the DoF. But by the time you select another point that dancer may have moved again! At best by using this technique you increase the time between shots each time you have to change the focus point. On the other hand, if Single AF is selected you can select a focus point and press the back button to lock focus when the dancer is in the middle of the DoF you desire. As long as you hold down the back button you can shoot as many frames as you desire without having to refocus or change the framing every time a dancer moves.
Beyond specific situations, one advantage of S-AF is that once focus has been achieved it remains locked and doesn't continue to search. When using C-AF/AI Servo AF it can be frustrating if the focus system attempts to adjust or refine the focus just as you press the shutter release. In S-AF once the focus is locked the focus system also stops consuming battery power. Itai covers these aspects very well in his answer.
For a more Canon-centric look at the same question, please see this answer to Is there any reason you would use one-shot focus over AI-Servo?