A "closed network" means that the SSID is not included in the beacon frames. Beacon frames are still sent periodically because they're important for the operation of the network, so you won't see any changes in the amount of traffic or in any performance aspect of the operation of an 802.11 wireless network.
From a security standpoint, the SSID will still be visible in probe requests and association requests, so it's not much of a security feature. But because you're talking about using it for your network admins, I think it's a decent usability choice--you are keeping your normal users from seeing one more option in their list of wireless networks that they could associate to.
As others have said, one downside is that people have to remember exactly how to type it. With older version of Windows and some driver software, if you type in the name of an SSID that doesn't exist, you'll end up creating an ad-hoc network with the mistyped name that other users might try to join, which gets you back to having end users who are trying to join the wrong network.
I suppose some older drivers could have trouble associating to networks with hidden SSIDs, but that "feature" of 802.11 is really old, so I would be surprised if it actually caused connection problems.
Each additional BSSID you add (which is how most modern APs and controllers create new SSIDs) will add some amount of overhead in beacons and management traffic to your network regardless of whether the SSID is advertised or not. If you can accomplish some other way of giving admins the extra access they need (RADIUS-based VLAN assignment, role-based policy enforcement on the controller, etc.) without creating a new SSID, you might think about that. But there probably won't be much difference in practice.