At a simple level, the difference is which method is called. The == method will attempt ot bind to operator== if defined for the types in question. If no == is found for value types it will do a value comparison and for reference types it will do a reference comparison. A .Equals call will do a virtual dispatch on the .Equals method.
As to what the particular methods do, it's all in the code. Users can define / override these methods and do anything they please. Ideally this methods should be equivalent (sorry for the pun) and have the same output but it is not always the case.
