Multimeter measurement
Is the circuit operating properly in both cases?

Probably the circuit is drawing current in short bursts.

When you have the meter set to the lower current range there is enough drop across the meter to affect the operation of the circuit or i overloads the meter such that it does not average the current correctly.

In those conditions I would put a large capacitor (several hundred or thousands of microfarad) across the supply rails of the circuit to slow down the meter response so it can average the current correctly and the capacitor supplies the current during the pulses.

Typically a DVM has a voltage drop at full scale of 200-500mV so on the 20mA range the meter will have an effective resistance of 10-25Ohms.  On the 200mA range it will be 1/10 of that.  This extra resistance on the supply rail may affect the circuit.