Two possibilities come to mind:
1) The first device had been set to use a static IP address on your Wifi network, and that got copied to the second device when you restored the backup to it. Fix would be to change the fixed IP address on the second device, or use DHCP.
2) Another possibility (though I don't know if this would actually happen) is that the second device ended up with the same MAC address as the first device following the restore. That would mean that the router would not be able to differentiate between the devices and it would just keep handing out the same IP address whenever either device asked for one. If that does turn out to be the case then I think you might have to take the second device back to its stock ROM and then re-install CM10.1.3 from scratch, rather than by restoring a backup from the first device.