You have to remember what account you have signed in to (to start MC). If you've been to the physical console on the computer, you might have to use /console when connecting with RDP. The best thing you can do is to start MC in the remote session that you normally log in to. Be it a normal user or administrator. As other mentioned, it's important that the RDP settings is set to play the music where it's supposed to play. If a client is connecting to the server, it should not matter much where it's played I think. Because the output should go to the client anyway. But if the server outputs the music, then you have to set the correct options.
If you have to use the Console admin to start MC, then you can use the following command to connect to console.
"mstsc /v:machineName /console"
This does however, have no affect on Win7, 2008 Servers and above I think.
If none of this work, I think I'm out of ideas at this point.