First of all, i'm not an expert on this..
Even though you have two monitors, there's only one desktop, spread across the two monitors. I'm guessing the external monitor is to the right of the notebook (at least in configuration). If you configure the external monitor to the left of the notebook, your icons will probably show up on the external display, with the panel still on the notebook. If you display the same image on both monitors, well, then you'll have the same image on both, but then of course you can't have different resolutions. I have a dual monitor setup (one monitor and one flat screen tv), however i have Nvidia graphics and can configure the two as separate X-screens in the Nvidia configuration tool. I'm then able to have individual panels and icons on each screen, with the disadvantage of not being able to move windows between screens. I have tried to configure a dual-screen setup on a computer without Nvidia graphics as separate X-screens without any luck..