Put simply, Irish republicans were not interested in a semblance of equality. In 1973, they were engaged in a paramilitary/terrorist campaign, focussed on removing entirely the British presence in Northern Ireland. The goal (in theory at least) was a united, 32 county socialist republic in Ireland, with no British involvement, and the strategy that they had adopted was an armed campaign. Political engagement was not on the agenda at that point, and wasn't really until 1982. It's been retconned as a struggle for equality more recently, but that wasn't the goal at the time.

By contrast, nationalists, in the form of the SDLP, were interested in equality (they were, after all, the party of civil rights). Consequently, they took part in the elections and formed part of the short-lived executive until its collapse during the Ulster Workers Council strike.

As another commentator said, the IRA were focused on a solely military strategy until the early 1980s. The IRA considered itself the only legitimate representation of the Irish state. Both the 26 county Republic of Ireland and Northern Ireland were British creations which were illegitimate and unreformable. IRA and Sinn Féin members were not to stand for or take seats in any assemblies in these entities.

There was considerable suspicion of political activity among the rank and file who had joined the Provisional IRA in part because they thought the Official IRA were too political. They also opposed the concept of consent to a united Ireland which they characterised as a unionist veto.

The first questioning of this approach began among prisoners in Long Kesh in the mid 1970s. In particular the "Brownie Letters" written by Gerry Adams discussed the need for street level politics (though not elections). This didn't really take hold until the Hunger Strikes when Republican candidates contested elections but didn't take their seats led to the so called 'ballot box' and armalite strategy. Even the decision to take seats in the Dáil caused a split.

Despite this strategy the political wing was firmly subordinate to the military until at least the 1990s.

There is some good reading on this. Richard English Armed Struggle is the best single history and Henry Patterson The politics of illusion is a critical look at Republican ideology.