This pattern is the first of the first of the Structural Patterns in the G04 book, and for the most part, it is meant to be used solely when you are integrating two or more external systems / libraries. When I saw it used inside a system, it almost always a Bad Thing, mostly because inside a single system, you want to use one of the other structural patterns.
HttpContextWrapper is a good example of using this pattern, and it links back nicely to the previous discussion on Singletons. HttpContext.Current was an issue, because it didn’t allow easy overriding / mocking / testing. The new HttpContextBase class was introduced, but due to backward compatibility concerns, HttpContext’s base class could not be changed. The solution, introduce HttpContextWrapper implementation that adapts between the two.
Useful pattern, but all too often people try to use it to abstract things, and that is most certainly not what it is meant for.
Recommendation: Use when you need to integrate with code that you can’t change (3rd party, legacy, compatibility concerns) using a common interface, but avoid otherwise.