From here:
Adapter provides a different interface to its subject. Proxy provides the same interface.
You might think of an Adapter as something that should make one thing fit to another that is incompatible if connected directly. When you travel abroad, for example, and need an electrical outlet adapter.
Now a Proxy is an object of the same interface, and possibly the same base class (or a subclass). It only "pretends" to be (and behaves like) the actual object, but instead forwards the actual behavior (calculations, processing, data access, etc.) to an underlying, referenced object.
Extrapolating to the electrical analogy, it would be OK that the use of an adapter is visible to the client - that is, the client "knows" an adapter is being used - while the use of a proxy might more often be hidden, or "transparent" - the client thinks an actual object is being used, but it is only a proxy.