Adapter pattern provides a different interface to an object. But the Proxy always provides same interface to the object.
Adapter is like providing an interface suitable to client’s use. But Proxy is same interface that has additional feature or check.
E.g. In electrical appliances we use Adapter to convert from one type of socket to another type of socket. In case of proxy, we have a plug with built-in surge protector. The interface for plug and the original device remains same.