Webservice is used to host our applications(software or 
programs) over Internet, so that others can use them or the 
Clients of Our Website can use them.

whereas Remoting is used to Connect remote systems or 
server in order to Communicate Online or Offline depending 
on the Connectivity. ( eg:  Yahoo Messenger where we can 
Chat Online or send Offline messages to Our friends or 
Contacts)

Both webservices and remoting support developing 
distributed application and application integration but you 
need to consider how they differ before choosing one 
implementation on other. 

For the application that requires interoperability 
(services developed in multiple platforms to interact) - 
best bet is web services. 

If performance is key priority for your application then 
you can use Remoting.

Both Webservices and Remoting are used to developing 
distrubuting Application. Among these, 

      WebServices are provide Functionality other Platforms 
like Unix,Linex. Means  Webservices are return data in from 
of XML Formate. This is the Universal formate. All 
Languages supports XML. So Webservices is better, When 
these are Communicating othere Technologies.

       Remoting are provide Functionality for Sampe 
Application.Means .NET Application. These are not Suppoting 
Other Platforms.