The resistor is there to limit the amount of current that can flow through the circuit.
Think of a valve on a water hose. No matter where on the hose that valve is, it still limits the water flow.
It's just like if you had a switch on the LED circuit. Whether the switch is on the anode side of the LED or the cathode side, the light will still turn off when you open the switch because then the charge can't move through the circuit.