Windows 8 Resolving Realtek NIC Driver Issues When Swapping Motherboards with Win 8.1

The identification you are seeing refers to the hardware. Since one is 8111E and the other is 8111G, it is correctly identifying different hardware. I don't know if they are supposed to use different drivers. It is possible that both use the same driver, or it is just using the driver you have on the HDD, which is correct for one but not the other.

Windows installs on the HDD, not the motherboard. However, the motherboard is the primary thing that defines the Windows license. If you keep moving the drive to different motherboards, you are likely to run into problems with the license, particularly after 30 days from the original installation. You can do a brief, temporary swap, like to diagnose a hardware problem, but you will want to put the drive back with the motherboard that was originally used when Windows was installed.

Part of the Windows installation is loading the correct drivers for the hardware on that machine. If you move the HDD to another computer or motherboard, some of the installed drivers may not be correct for that hardware, which could be what you are seeing. Generally, the process should be to assemble all of the hardware, including the HDD for that machine, and then install Windows. That way, the proper drivers will get loaded and the Windows license stays associated with the hardware.