What are the differences between ADS + INS and ADIRS / ADIRU?
The primary difference is packaging, which saves space and weight. The ARINC 704A IRU is a 4 MCU size LRU. Likewise, the ARINC 706 ADC is also a 4 MCU unit. The ARINC 738A ADIRS combines both functions into a single 4 MCU LRU. Though the ADIRS also includes remotely mounted Air Data Modules (ADM) which reduce the amount of pitot/static plumbing.  

Functionally, the ADIRS performs the same functions as the separate IRS and ADS. A big drawback to the older ADS was the need to run the pitot and static lines all the way to the LRU. These lines are a maintenance headache, requiring multiple connections all of which have the potential to leak.  

The ADMs can be mounted near the pitot and static sources significantly reducing the plumbing. The ADM converts the pressures to digital signals passed to the ADIRU via ARINC 429 or similar data bus. This simplifies installation, maintenance, and weighs less.

It should be noted that newer aircraft are moving to even newer architectures. The Boeing 787 dispensed with ADIRS and went with the Goodrich SmartProbes for air data and an Earth Reference System (ERS) to replace the IRU.  

The Goodrich SmartProbes do away with pitot/static plumbing entirely by incorporating the pressure conversion in the probe base. They also electrically cross-connect the probes to perform the ADC function with no separate LRU.  

The ERS addresses the FAA/EASA concerns over common mode faults in the IRU by using an IRU/AHRS mixed sensor architecture. In each of the two ERS, the IRU acts as the primary reference with the AHRS acting as a monitor. The key is that the IRU and AHRS use different hardware and software so that a common design flaw can't affect both and cause an undetected failure.  

It's just evolution in the function of the aircraft sensors. They do what they need to do. The big demand from Boeing, Airbus, et al, is to reduce size, weight, power consumption, and cost.