Accuracy when measuring current (mAmps) and small voltages
Your multimeters will have round about 1 or 2 ohms input impedance. They have a small (but not zero) impedance so they can develop a voltage across it which can be measured. In other words they "infer" current by the voltage developed across the small resistor and this small resistor is affecting the current through the LED.

Both meters will have slightly different resistances and hence the different readings.