This seems to happen on Windows 7 (at least, it did for me).
When I first plug in, the SD Card mounts automatically. In order to access the phone storage on Win 7, I need to wake the phone, swipe down the windowblind, click the listing for Disk Drive, confirm the Disk Drive selection, then click done.
After that, the internal storage mounts. On my Windows boxes it is just labeled "removable storage" -- Don't look for the drive letters mentioned by I am138 -- every computer has a different number of drives attached and you probably won't see the same drive letters that he does (they are G: and F: on my machine).