Look at the kernel modules loaded for the computer that works. (lsmod)
Also look at the hardware detected by the kernel. This command will filter out the other devices:
/sbin/lspci -v | sed -n '/Ethernet/,/^$/p'
03:09.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL-8169 Gigabit Ethernet (rev 10)
Subsystem: Netgear Device 311a
Flags: bus master, 66MHz, medium devsel, latency 64, IRQ 17
I/O ports at ac00 [size=256]
Memory at fddfe000 (32-bit, non-prefetchable) [size=256]
Expansion ROM at fdc00000 [disabled] [size=128K]
Capabilities: <access denied>
Kernel driver in use: r8169
Kernel modules: r8169
Since one computer has a cloned drive, make sure that you change the hostname and if you use static addressing, the IP address. If using dhcpd then check that there is a unique identifier in /var/lib/dhcpcd/.
Also try running the network setup. You may need to delete the nic device in the devices setup and start over. The device may have been given a unique hardware identifier and this information may be wrong if it is based on or contains the MAC address.