What are the differences between proxy, wrapper or a façade classes
They all seem to be the same to me, they take an implementation, encapsulate it and then methods are
AFAIK there is not a pattern called wrapper. Seems more like a generic definition of one of the behavioral patterns.
Facade
Do not wrap a single class but several. The intent is to make a complex API easier to work with. A .NET example is the WebClient
class which uses HttpWebRequest
/HttpWebResponse
Proxy
Proxies acts as the original object but contains additional logic such as handling a connection or cleaning up resources when done.
If you are using WCF you can generate a client proxy by supplying the WCF service interface.
Additional patterns
There are some more patterns which also is "wrappers":