Hi,
Is the PC connected directly to the LAN port of the router or via the ethernet socket on the wall?
Try unplugging the network cable from the PC and plug it into the laptop, see if the laptop behaves the same or not.
On the desktop PC, go to Device Manager, expand network adapters, select your network adapter in use, click on the Power Management tab, uncheck the checkbox: "Allow this computer to turn off this device to save power". Click on the Advanced tab and disable any options regarding energy saving: Energy Efficient Ethernet (EEE), Green Ethernet etc.
Specify the exact model of your ethernet network adapter.