turning off an ATX-PSU
An ATX motherboard connects PWR_ON# (green wire) to ground to turn on power and disconnects it from ground to turn power off. All power (except stand by 5V) is controlled by that.

If you want separate push buttons for on and off, you’ll need to make a latching circuit, probably powered by the SB5V, to pull the green wire to ground. Your ON button triggers the circuit. The OFF button could be a normally closed button to cut power to the circuit or a NO button to trigger a reset to the circuit. This circuit could be as simple as a DPST 5V relay where the second poll hold power to the relay, or an SR flip flop (Wikipedia) made from a couple of BP transistors.