In short: the 24 bit address is critical on the technical layer of Air Traffic Control, but it is not used in the operational level. For identification of aircraft and association to flight plans, the air traffic controllers and the flight data processing systems rely mostly on the Mode A code and the aircraft ID (call sign) transmitted by the transponder.
The ICAO 24-bit aircraft address is an essential element of the way Mode S radar and TCAS work.
Before Mode-S radar, only modes A and C were used by civil ATC secondary radars.
Mode A and Mode C are unaddressed; every transponder receiving a Mode A or Mode C interrogation will reply*. This causes two problems, garbling and FRUIT.
Garbling is caused by the overlap of replies from multiple transponders. A transponder replies on 1090 MHz in the from of a number of pulses. For Mode A & C, there are at most 12 pulses between a set of framing pulses. When multiple aircraft reply to the same interrogation, these pulses mix up and it becomes difficult to find out which transponder sent which pulse. This leads to false altitudes or false squawk codes.
FRUIT (False Replies Unsynchronised to Interrogator Transmission) is caused by replies that are triggered by other radars. When multiple radars operate in an area, transponders can be quite busy and the radar is easily mislead by a reply to another radar. In addition to the timing (range) being wrong, the reply can be Mode A (squawk code) while the radar interrogated Mode C (altitude). There is no way to see what kind (A or C) reply was sent.
To overcome these problems of Mode A/C, Mode S was introduced. Mode S is quite different from Mode A/C and requires a more sophisticated radar and transponder. It uses the 24 bit address to distinguish responses from various aircraft and to interrogate specific aircraft.
Most Mode S interrogations are addressed; the interrogation contains the unique 24 bit address of the aircraft it is interrogating. This reduces the probability of garbling since other aircraft will not reply
Unaddressed Mode S interrogations will solicit replies that contain the address of interrogator (radar). This prevents FRUIT since the radar can verify that the reply is correctly addressed.
Unlike Mode A/C which does not have any error checking, a Mode S reply contains a 24 bit CRC check code.
Mode S Altitude and identity (squawk) replies have their own identifier so they can be distinguished.
Mode S radars can be operated in clusters and coordinate between them who is interrogating which aircraft at which time. This further reduces the message load and garbling / fruit probability.
For TCAS, the Mode S address is used to distinguish replies for various aircraft in the vicinity of the own aircraft. The TCAS transponder interrogates omnidirectionally (in all directions) and get's replies from all direction. Again, this is very susceptible to Garbling and FRUIT. Whilst TCAS can work with Mode A/C, for its reliability and coordination of resolution advisories, the 24-bit address is crucial.