I've never done any programming with .Net Remoting but I think the concept is similar to RMI/CORBA in Java where remote invocation of a remote object is carried out. So .NET remoting is more for binary communication.
In the question, it refers to .NET remoting using SOAP. I think (not totally sure) that it is equivalent with ASMX in this case. However, ASXM provides in addition a well defined wsdl and Document/Literal style where as .NET remoting is more OO suitable.