I'm no expert on Ireland, but a little bit of history is relevant:
the Ulster Transport Authority (thus called!) drastically thinned out rail services in NI in the 50's and became (if it wasn't already) very bus-orientated, to the point that services to Derry and Dublin survived for political reasons as much as anything.
Indeed, although any analysis cannot really get away from seeing that the rail closures were political (or some might say 'sectarian') too: post WWII, Northern Ireland was controlled by a regional Unionist government in Stormont, that was extremely partisan it dealt with Protestant versus Catholic communities. Any infrastructure in 'catholic areas' was generally run down, and particularly any infrastructure which could be seen as part of a single-island-of-Ireland: so any cross-border regional lines were for the chop.
The Irish govt in Dublin protested but only managed to get the Dublin - Belfast line kept open.
The Stormont govt also closed the remaining canals, and put all their cash into motorways (motorways which, again, went to places where protestants lived, not to where catholics lived... and *certainly* not across the border to the Republic!)
As everyone will no doubt be well aware, the Stormont government's sectarian policies led to widespread civil unrest by the end of the 1960s, mushrooming into 3 more decades of terrorist violence (and several train lines blown up!)
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Really, you have to understand that in NI, everyone who can, drives. Driving is the most common form of transport. It might be a vicious circle... public transport is not as good so everyone drives, so public transport use isn't as high. In some more rural parts of the country, especially out in the west where the rail network doesn't reach, really the car is the only method of getting about. .
It is *totally* a vicious circle. But it is also daft, as in the end, road transport corridors into Belfast became so clogged that, my the start of the 1990s, I remember it taking over 1½ hours to get home from school on the bus (6 miles from Belfast city centre south to Carryduff) nose to tail traffic the entire way.... it had got to the point where it was quicker to walk!!
The most egregious flaw in the transport policy to my mind is the lack rail links to the airports. There is NO station at Belfast International, and indeed they closed the Antrim-Lisburn line that goes past it! And there's little more than a halt at Belfast City, with very poor links for foot passengers.
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I think you have to be aware of the size of the network, and the population (and thus the demand) it is serving, both of which are of nothing like the scale you have on the mainland.
You also have to be aware of how that population is distributed, and why they might be travelling in the first place. People commute to work (although most of Northern Ireland's jobs are in the public sector), and NornIrond still has the 11+ and grammar/secondary schools, with many thousands of kids commuting into big grammar schools in Belfast.
About 1/3rd of the population lives in the greater Belfast area (including ribbon development along the north and south foreshores of Belfast Lough - Banger, Newtonabbey, Carrickfergus... - and settlement further inland along the Lagan valley - Lisburn). These all act as dormitory towns for people commuting to work/study in Belfast.
There are indeed rail lines along both Lough foreshores and down into the Lagan Valley, so people can and do use them to commute.
However, there are NO rail lines AT ALL any more in counties Armagh, Fermanagh or Tyrone! So obviously none of their populations can commute anywhere by rail.
Commuting to (London/)Derry, and places in Antrim (Coleraine, Larne) is possible but people living there are less likely to commute to Belfast as there are some modicum of jobs in the aforementioned places, they are not really dormitory towns.
Partitioning Ireland meant that many of the towns and cities in the west of the north (Armagh, Derry, Enniskillen, Strabane, etc) that up until then had been market centres and places of employment had half their catchment areas (from counties Monaghan, Cavan, Leitrim and Donegal) lobotomised, and their rail lines scrapped. So people's reasons for travelling were changed.
Another reason for travel is of course for leisure and holiday, often involving airports - but as mentioned above, NI's airports don't have rail links (!)