Does this cable connect to the ATX_12V_2X4 connector of my motherboard? I'm referring to the connector infront of the SPDIF connection of my motherboard.
Yes. In addition to all the Molex, SATA plugs and PCI-E plugs, you should have 1 24 pin, 1 4 pin and 1 8 pin. You will use the 8 pin since your board supports it.
Plug the connector in your photo into the 8 pin connection the motherboard.
One of the two PCI-e power cables that comes with the Earthwatts 650 has a 6+2 pin power connector for a total of 8 pins. That one is for video cards. It can be used as either a 6 pin or an 8 pin connection.