Motherboard fan controllers are built in to the motherboard; you use software from the motherboard site to control the fans.
External fan controllers with knobs connect to the fans; there's no motherboard connection except possibly USB so you can manage by software. Depends on the controller.
The Corsair Graphite 600T is a case. The fans can be controlled by most motherboards.
*Fans are either Voltage or PWM. My Asus motherboard can control only PWM fans for the CPU, but it can control either Voltage or PWM fans via the fan connections on the motherboard.
Example:
1) Connect Case Fan to FAN_1 (case fan) connection on motherboard
2) Ensure BIOS has fan control enabled
3) Download and run fan control software from motherboard site, and adjust fan profile as desired (such as 40% until 40degC then ramp up to 100% at 80degC).
Other:
I believe the Corsair water-coolers for CPU's like the H80i etc use a USB connection to manage the fan profile with no requirement for the motherboard CPU fan connector.