Constant current - what is the difference between loads
The difference is how the load draws current.
A 12V 600mA LED would ideally draw 600mA only when it has 12V.
It will draw less current when it has less voltage, for example it might draw 500mA at 11V.
So for example if you power it with a power supply that is set to 12V constant voltage with 500mA current limit, the LED would initially try to draw 600mA because of the 12V provided, but the power supply hits the current limit and it will drop the output voltage until the current draw is 500mA and the power supply output voltage ends up being 11V.
The dummy load is most likely using constant current mode. If you set it to 400mA, and connect it to 12V power supply with 500mA current limit, the dummy load will start drawing 400mA, and since the power supply has no problems providing 12V at 400mA, the voltage will stay at 12V.
When you start cranking up the current on the dummy load, you can hit up to 500mA, because the power supply is set to provide only 500mA, and voltage is still 12V. When you set the dummy load to 501mA, it will try to always draw 501mA, no matter what. The power supply has now hit the 500mA limit, and will start dropping the output voltage to keep the output current limited to 500mA. But the dummy load would still like to keep drawing 501mA regardless of the voltage, so even if the power supply output drops to 1V, the dummy load would still try to draw 501mA and power supply must continue dropping voltage in order to try limit the current, and the result is that power supply output voltage will be quite low near 0V and much below 501mA flows as there is not enough voltage in the system for the dummy load to draw 501mA at so low voltage.
Another example with e.g. small 12V motorcycle battery. Assume it empty at 8V, and needs charging to 12V, and it must be charged at maximum current of 500mA so it does not explode. So you set the power supply to 12V voltage, and current limit to 500mA. As current is limited to 500mA, the power supply voltage is also 8V. When battery charges at 500mA constant current, voltage rises up to 12V, and then power supply gets limited to 12V. The battery won't draw 500mA any more, but current drops more and more when battery gets more full. If left on indefinitely, the battery would reach 12V too, and since both battery and supply are at 12V, and current would drop to 0.