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You've probably done most of the things I could think of which does suggest something might be wrong with a hardware configuration, possibly a duff driver. You are posting this which shows your old computer connection works, so that rules out the TW connnection and modem.
You say you have reformatted the PC - that would suggest that it's going to be free of virus. I just wonder if the drivers have reloaded properly? It is allways possible that something hardware has broken.
I don't want to teach you to suck eggs but need to check that you do power off the modem, and unplug the old computer network connection, plug in the new PC network connection, power on the modem, give that time to settle, before booting up the PC.
Ethernet is generally a more reliable connection method. Do you use ethernet or USB for your old PC?
Have a look at your windows security settings (Control Panel). Start off making sure there is no firewall enabled, and don't install Norton, or enable any other antivirus trial software that may have been supplied with the PC.