Adapter provides a different interface to its subject. Proxy provides the same interface. Decorator provides an enhanced interface. Adapter is meant to change the interface of an existing object. Decorator enhances another object without changing its interface.
Decorator is thus more transparent to the application than an adapter is.
As a consequence, Decorator supports recursive composition, which isn't possible with pure Adapters. Remember that Adapter makes two existing interfaces work together as opposed to defining an entirely new one.