Are the computer using the same computer name? Did you use a restore disc on both systems? or the one from the computer that did not need a restore on the second one that did. If you did that, they may have the same name. You generally should not use a restore disc on more then one system. Each has its own disc and product key.
Can you check and make sure they are set to obtain an IP automatically.
If both systems have the same name, or IP addressed manually entered they will conflict.
Right click My Computer or Computer on Vista/7. Click on the Computer name tab or click Advanced system settings on the left hand pane then use the computer name Tab in Vista/7.
If they are the same, chance one and reboot.
If its not that, make sure the systems are not running the same IP. Start - > Run -> type cmd and hit enter, if Vista/7 does not have the run option just type cmd in start search.
A command prompt will pop up. Type ipconfig /all and hit enter. You should get your ip address. It will look something like 192.168.x.xxx.
If they DO match, post again and I will tell you how to manually set a different ip or set it to get one from the router.