While I understand the 'same cable' works on the laptop, at this point it is looking hardware wise. I would try with a different cable, then try with a different port. If this is a OLD (I would suggest it is with the 10/100Mb port on the router) device it may not understand how to change to meet a 1Gb NIC (seen a few happen like this back when). I would then suggest if you replaced the cable, tried a different port, power cycled both hardware (PC and Router) then they may just need to issue a up to date one.
Now normally the 'industry standard' is a all in one device, Modem, Router, Multi LAN port and Wifi built in. If you do not have such a device from the ISP then I would HIGHLY recommend swapping it out with the 'current model'. You can validate that by checking (without being logged in) the ISp website and see what they offer "new customers" and what model router they offer. Alot of people don't realize they can get a updated router without question. I went through three just because they went from the 30Mbps cap to the 75Mbps, then doubled to 125 only to now offer up to 300Mbps I believe, which I had been issued two newer routers after just getting the first one when I moved here because of the changes and now had a recycled Business Class Cable Modem/Router Wifi for our family.