P mode can be used as a modified Av mode, in that you can still override the aperture setting and also compensate. Used this way it can be useful. So it's not as much for 'losers' as you might at first think.
Help me out here - why wouldn't I just use Av mode? I mean, assuming I'm not Ken "The P stands for Professional!" Rockwell.
Fair question; PBD has covered much in his post above. P allows you to operate broadly as Av, taking control of aperture and iso with a variation on EC. However P will revert back to a 'middle ground' ss/ap balance as soon as the camera is idle. Depending on your situation this may or may not be an advantage in a swiftly changing environment such as parts of a wedding - except when you put a flash on the camera you loose this control. I'm not sure how someone shooting a wedding in P mode deals with this.
The one mode I have never used on digital is Tv. In the film days when you had a certain ISO loaded you might want to have the ss locked. However since digital and immediate access to ISO variation I find it is much better to ensure the ISO is suitable to allow the ss I require at the appropriate aperture. DoF is always important, shallow or deep depending upon the result you want, so to be quite honest I personally have more use for P mode in a modern camera than Tv.
As unfocused says, green box is useful when handing a customised camera to someone to use. If I hand my camera to my wife I put it in green box mode.
For myself, with the exception of shooting panoramics, by far the most used mode is Av. I even use Av when working from a hand held incident light meter reading. Set the required aperture then compensate to shoot at the incident meter reading. If I am trying to nail an exposure and 'picture style' in camera when shooting around 500 images at an event, this saves so
much time in post.