The important thing when swapping over the Cable Modem connection from PC to Router is that the Cable Modem *MUST* be powered off and left without power for two or three minutes. This is so that it "releases" the association between the existing connection and the IP address it has allocated. It will then recognise a new device has been connected and it will allocate a new IP address.
Power on the modem and wait for it to be fully booted and online. Power on the router and wait until it is fully up and running. Then boot the PC.