If you implement explicitly, you will only be able to reference the interface members through a reference that is of the type of the interface. A reference that is the type of the implementing class will not expose those interface members.
If your implementing class is not public, except for the method used to create the class (which could be a factory or IoC container), and except for the interface methods (of course), then I don't see any advantage to explicitly implementing interfaces.
Otherwise, explicitly implementing interfaces makes sure that references to your concrete implementing class are not used, allowing you to change that implementation at a later time. "Makes sure", I suppose, is the "advantage". A well-factored implementation can accomplish this without explicit implementation.
The disadvantage, in my opinion, is that you will find yourself casting types to/from the interface in the implementation code that does have access to non-public members.
Like many things, the advantage is the disadvantage (and vice-versa). Explicitly implementing interfaces will ensure that your concrete class implementation code is not exposed.