If you have a Silverlight Application hosted in an AppPart cannot call WCF service in the same SharePoint 2013 site, but it works if it is in a WebPart in the same site, you will need to look the following:

1. Install the latest SharePoint Server 2013 cumulative update (if you are using Sharepoint Server 2013 RTM, you are likely to see this issue)

2. Add the AppPart domain URL (for example: http://apps-c8924ecab91592.apps/) into the "Local Intranet" security zone in IE.

 

When you experience the issue, there is no error message. However, if you capture first chance exception dump using adplus or DebugDiag, you will see System.UnauthorizedAccessException with Exception Message:

"Debugging resource strings are unavailable. Often the key and arguments provide sufficient information to diagnose the problem. See http://go.microsoft.com/fwlink/?linkid=106663&Version=5.0.10411.00&File=mscorlib.dll&Key=Arg_UnauthorizedAccessException
" if you analyze dumps in windbg.

The two steps above resolved the issue.