Hi! I've encountered this problem as well. I don't know why exactly, but certain variations of the 12" 1.33GHZ iBook require a very specific screen, namely the ChiMei N121X5-L03 (pictured). I ran across about 75 of this laptop a year ago, about half with this issue, and mostly with broken or missing screens, and it took a while to come to this realization, but that was definitely the problem -- use the right screen, and they work fine, and use the wrong screen (any screen other than that one), and you get a black screen and a very hot video cable right by the plug. If you look at your old screen, I'd be willing to bet it's a ChiMei.
The "problem" is with the cabling (the video cable, if I remember right), and not the laptop specifically. If you replace the cabling with that of another laptop that doesn't need this specific screen, you can then use any screen on the laptop.
But again, I don't know what it is about the ChiMei screen and the specific cabling in these laptops that makes them work together. The ChiMei screen DOES work with any cabling, which means it's the screen cabling that has the specific preference for the screen, and not the other way around. It's the only instance of iBook 12" screen incompatibility I've encountered. I mentioned all this about 6 months ago, but no one really seemed to believe me or confirm another case of this at the time, so it's good to hear others have found the issue as well.
Also, I just uploaded a second picture, which shows the back of the screen, because the ChiMei screen has a distinct appearance along the back that is helpful in identifying it. These can be purchased on eBay by searching for the part number, however the prices are not generally very reasonable, so swapping the cabling is likely the cheaper way to go.
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Can you provide us with the model numbers of both units? Ralph