I think it's important to make that distinction here because "Ireland" can also sometimes refer to the whole island.
From article 4 of the Constitution: "The name of the State is Éire, or, in the English language, Ireland."
https://en.wikipedia.org/wiki/Names_of_the_Irish_state
http://www.irishstatutebook.ie/eli/1948/act/22/section/2/ena...
1: https://www.gov.ie/en/publication/d5bd8c-constitution-of-ire...
Using "Republic of Ireland" is useful when distinguishing between the island and the state.
The one Irish national entity to use the name Republic of Ireland is the football team, which does so as a result of a FIFA ruling as the Northern Ireland football team was the defacto successor of the pre independence football team so kept the name Ireland until FIFA insisted on the disambiguating names.
Look, in a informal context nobody is going to bite your head off for getting it wrong. It's not at the level of going around Zimbabwe talking about Rhodesia, both due to the lower phonetic distance and subsequent outcomes for both countries after British rule, but it comes from the same place. Which is why just this week the normally soft spoken President of Ireland was issuing a rebuke to the DUP for rendering his title as "President of the Republic of Ireland" in the context of a diplomatic dispute. Because it's just not the name of the country.