.NET remoting is specifically used when you are using only 
Microsoft platform. Remoting cannot be used between .net 
client and non .NET clients.
Web services though can be used for communication on 
MIcrosoft technologies, but their main feature is Platform 
Interoperability.
So if you are using client and server supporting .NET 
framwork,  iwould recommend to use .NET remoting because 
you can customise it to great detail and also its faster.
If you are developing for cross platform technologies, you 
cannot use .NET remoting. So the only option you have is ot 
use web services

WebService is Platform,Object ,Programming Language
indepedent while remoting depends on platform means client
to be built using .NET Framework or other framework that
supports .NET Remoting
webservice can be accessed on http protocol and soap based
message formating while remoting supports tcp/http with
binary and soap based formating.
webservice is based on singlecall method means for every
request remote object is loaded, while remoting supports
singlecall and singleton method.
webservice lies under unbrella of Remoting.