There are two distinct Remote Desktop apps available from Microsoft for Windows 8. The desktop version of the app has all the same features as the Windows 7 version (as well as significant behind-the-scenes enhancements) and should be able to copy/paste files over the connection, as well as map a drive to a shared folder or mount the drives on either end, just like in Windows 7.
The is also an RT version of the app. This version has the same file system restrictions as other Modern UI style apps, and so it is much more limited for things it can do on the file system. This app was built-in for the Consumer and Release Preview versions of Windows 8, but now must be downloaded from the Windows Store.
Additionally, if you have a Windows RT tablet, you can still access the full desktop RDP client! Just hit
Windows Key +
R, and then type
mstsc into the box that opens, and you'll get the full RDP client.