IRS/ADC systems use a separate IRU and an ADC which is connected to lots of pipework so that pitot and static pressure can be sensed.
ADIRS's have a combined ADC/IRS and is fed pitot and static pressure from several Air Data Modules ADM's (1 pitot and 2 static ADM's per system). The main advantage is the removal of pitot and static pipework which can be prone to damage, blockage, water accumulation and it can be very labour intensive finding the location of these faults when they have occured.