The most likely advantage you'd derive from owning one is the slight reduction in scrutiny you'd receive in some countries because of the (lack of) negative political and historical connotations Ireland has due to its short history as an independent nation. In the vast majority of situations however, it would probably present no real difference.
There are also situational advantages such as being able to apply to intra- or international schemes as an Irish citizen if it suits, being able to claim grants for an Irish third level education, and apply to some unique Visas only available to Irish citizens due to treaty (there is one for the United States, if I recall correctly). But these are more benefits of citizenship rather than the passport, and often stipulate reasonable proof of residence.