It would be helpful if their website gave the background and history of the Society, but this is how I understand it and I am open to correction.
The 1963(?) Companies Act brought in the requirement for auditors of companies to be members of professional bodies. The recognized bodies at the time were the Institute of Chartered Accountants and The Certified Association.
So what about the guys who were acting as auditors who were not qualified? Well they got an exemption to continue acting as auditors. These guys got together and formed the Institute of Certified Public Accountants. The CPA then started requiring new applicants to pass exams. These were considerably easier than ACCA or ACA but have got more difficult easier.
So in the eighties I think, the unqualified accountants formed the Insitute of Incorporated Public Accountants. Bizarrely, they managed to persuade Mary Harney to recognize them for audit purposes. Some of the other firms sought a High Court review of the decision on the grounds that their training was not adequate, but the High Court confirmed the Minister's decision. So they have the same rights and authority as chartered accountants and certified accountants.
From a practical point of view, most employers would not value the qualification as much as the others.
Brendan