Tell you what, this going to sound dumb but try it, it worked for me before. Shut down both computers, and unplug the cable modem (don't just turn the swtich off). Go in the other room and get a drink of something. Like reseting the computer in your car by pulling the negative terminal, some cable modems may have memory and remember the MAC address of the computer they were hooked to. They probably eventually update once in a while but this is a way to force it. After leaving it unpluged for a few seconds, plug the cable modem back in (to the new computer too) and let it go through it's diagnoistic light flashes. Once it's stable, turn your new computer on, hopefully the cable modem will see the computer and establish it's connection. If it doesn't, try the run/cmd then ipconfig /release ipconfig /renew.

What OS is on your old computer? It's possible your ISP doesn't use DHCP and your given a static address. If that doesn't work, we'll have to check the old computer for an IP.