I am pretty sure you are doing this, but wanted to make sure... You are resetting your modem when you connect your server right? Every cable modem that I have had will not allow another MAC address to connect unless it is reset. That would explain why the router connects, but the server doesn't. Like I said, I am guessing you are already doing this...
It seemed to me that when I was changing servers, and had a similar problem with the NIC swapping issue, that I had make sure the server was off, disconnect my LAN connection from the modem, unplug the modem for 20 sec. or so, plug the server into the modem, turn the modem back on, fire up the server, then log in as admin and test internet connection.
That last step isn't necessary, but I liked to verify that I had plugged in the right NIC (as they got swapped around) while I was standing there.
Ryan