First and foremost: to be truly sure about the answer I'm giving you I need the exact model of your machine. You're telling me it's a Macbook Core Duo, thus an A1181.
The iFixit user community needs the last four letters and numbers of your Macbook serial number to be sure about the machine. You can find the serial number in System Profiler, or on the original Package.
If you're sure is a Macbook Core Duo, you should just connect the new topcase you've got to your logic board, without any other hassle. It will work. You can check this guide ( MacBook Core Duo Upper Case Replacement ) for references, both about installing the brown cabled ones, both about installing the silver cabled ones.
Actually, the Bronze Cable and the Silver Cable are different in the way they connect to the Top Case part, but they've got the same connector on the Logic Board part. I too used to have a Brown Cable, and now I've got a Silver.
Fact is that your model, the Macbook 1,1 and the earliest Core2Duo, the Macbook 2,1, came with a Topcase of a rather thin plastic that used to chip, bend, discolor and deform with wear and tear.
And that was the "v1" Topcase.
Because of all the complaining, Macbook started building sturdier, stronger topcases, the "v2", getting them from their now dealers, the Foxconn if I'm not mistaken.
Those studier topcases came with a Silver Ribbon, and were different in the way the ribbon latches on the Topcase, but not in the way the ribbon latches to the logic board
Then, came the v3 model, shipped with the latest Macbook A1181 (seemingly the one you got from iFixit), sturdy as the v2, but with the iTunes keys.
So, just connect it the way it is.
Furthemore, this is an help community staffed by happy customers, not by iFixit itself.
If your new Topcase develops issues, or you've got commercial questions, you should write a mail to iFixit itself.
But, if your machine is an A1181 (please, provide us with the serial number to be sure), there shoulnd't be issues.