Why does a Darlington pair have a higher input impedance compared to a regular normal BJT transistor?
The first transistor in the darlington pair has a  much lower  base current than the second transistor .Remember that the emitter current in the first transistor is the base current of the second transistor when the effects of any base emitter resistors  are small .This is usually the case when currents are high .The input current in the darlington is divided by the  current gain of the first transistor .The input voltage is 2 VBE or roughly double that of the straight BJT .For example if the first transistor has a current gain of 25  the input impedance goes up by about 50 .