The method of government formation differed in each.
In the Dáil Constitution, the head of government, the Príomh Áire/President of Dáil Éireann/President of the Republic was elected by the Dáil. He then formed a ministry.
Under Bunreacht na Éireann, the President of Ireland appoints the head of government, the Taoiseach, on the nomination of Dáil Éireann (the House of Representatives).