Currently, it seems that your device is configured in such a way that the default adb shell is not root, and you should use su to get the root shell. The DDMS file explorer in eclipse will not do this, so it will only have root access if adbd is configured to run as root and provides root access to clients by default.
You can check this from adb shell with
ps
(and see if the ps process is running as root or shell) or
getprop ro.secure
Using the su-root shell, you can remount the root file system and change ro.secure to 0 in default.prop - among other things, this will cause adbd to run as root and provide root shells by default. However, this has security implications, and you may not want to leave it that way. You will need to search the Internet for the exact command to reinstall the shell for your specific device (since for convenience, adb will not work if adbd is not yet root).
You can also see if the adb root command will work to put adbd in root mode.
Perhaps your system is not “rooted” in the sense of making any configuration changes, but instead only in the sense that there is the usual binary representation that allows unprivileged users who are aware of the possibility of obtaining the root shell - exchange tools would not know about this opportunity and therefore could not.