Translation:Republic of Ireland and Northern Ireland.
The state I live in is an artificially created one that is referred to here as 'Northern Ireland' most nationalists however refer to it as the north of Ireland and do not recognise the term Northern Ireland or the insinuation that its a separate country or nation from the rest of Ireland
Even in Ireland there is a difference between "Northern Ireland"/Tuaisceart Éireann and "the North of Ireland"/Tuaisceart na hÉireann. In fact, Irish people are usually the only people that understand the difference.
Most of the time you can use the familiar "the North of Ireland" or even "the North", but there are situations where the correct form "Northern Ireland" is required. One of the reasons for including both Tuaisceart Éireann and Tuaisceart na hÉireann in these exercises is to highlight that difference.
As for the GAA - leave out the full-stops!
The North seems to be currently favoured by most. An interesting sign of the times, to me at least, is the increasing tendency of English people in England to ask me if I'm from "The South", by which they mean the Republic of Ireland: being the helpful person I am I always answer "No, I'm from the North West"; which I am if we're going to get all geographical. Although being a 'Southerner' is an improvement on "Fenian B**d" though.
The Republic of Ireland will be playing Northern Ireland in Dublin on November 15th 2018.
Not "the North", or "the North of Ireland", just Northern Ireland.
There are a number of different informal ways of referring to either state that are more or less appropriate in different circumstances, but when used in conjunction with one another, they should match. So you can use informal shorthand like "Ireland, North and South" or "the Republic and the North" but if you're using the full technical description of one, you should do the same for the other. "The Republic of Ireland and Northern Ireland".
"The North of Ireland", "the North" and "Norn-Iron" are informal terms that are usually unobjectionable in informal situations. There are other terms that are favoured by different communities that should generally be avoided unless you know your audience. But you should understand the difference between formal and informal terms.
I'm Irish, we generally refer to the Republic of Ireland as "Ireland" or the "Republic", but I've never heard anyone say the "Irish Republic" unless they are referring to history, and Northern Ireland as "Northern Ireland" or "The North" (be careful using "The North" though, as it can refer to some of the northern counties that are in the Republic of Ireland). The whole Republic of Ireland and Northern Ireland situation is very delicate and routed in history, so you do have to be careful how you refer to it.
I'm not a native speaker, but I think they are indeed different. As you can see, "Republic of Ireland" (the country) is "Poblacht na hÉireann." However, if you want to say "Irish Republic" in the sense of "a republic that has Irish characteristics" or something, then you'd use the adjective for Irish, which is Éireannach. So you'd say "Poblacht Éireannach."
Worth noting, though, that http://www.focloir.ie/en/dictionary/ei/irish?q=irish lists ". . . na hÉireann" as an acceptable way of saying "Irish ...," so take this all with a grain of salt. I'm only answering because it's been three weeks since you asked. I'm not too sure anymore.
"Northern" is an adjective. Tuaisceart is a noun.
There really isn't a simple, one-line explanation for this difference, but both phrases are grammatically correct in their respective languages, and they are accurate translations of each other, even though the grammatical structure is different.
The Republic of Ireland is the official and legal "description" of the state (see The Republic of Ireland Act 1948). It is used when it it necessary to differentiate between the political and the geographical "Ireland".
The Irish Republic was the name used prior to the creation of the Irish Free State in 1922.
Since then, the phrase has been at times a politically loaded term favoured on the one hand by physical-force republicans who denied the legitimacy of the Irish state, and on the other hand by many UK based media operations and British politicians who couldn't be arsed figuring out the terminology, as well as by some Unionists (for any one of half a dozen reasons).
So yes, there is a very real distinction between these terms in contemporary usage within Ireland. Like the phrase "the British Isles" , the phrase "the Irish Republic" can cause minor irritation if used in ignorance, or some degree of offence if it is perceived that the usage is deliberate, even if you argue that those phrases are technically justified (for geographical or grammatical reasons).
No, it should not be an acceptable answer here, for reasons that have been detailed in this and other threads that use the term Tuaisceart Éireann. In informal speech, it often won't matter, but "the north of Ireland" is an ambiguous term - Donegal, for example is, geographically speaking, in the "north (small n) of Ireland", but is definitely not in "the North (big n) of Ireland" (which is, by definition, an informal term, as it has no technical or legal meaning - the legal name being "Northern Ireland").
It's like translating na Stáit Aontaithe as "America". It often doesn't matter, but that doesn't mean that it is always acceptable.