Northern Ireland has always been the UK's problem child owing to The Troubles. This was a period of intense political violence between, broadly speaking, Irish Catholics (often Nationalists, Republicans) and Ulster Protestants (often Unionists, Loyalists), who generally identify mutually exclusively as Irish or British respectively. This makes the prospect of a simple referendum completely unrealistic. It would unlikely change anything.
A good book to explain the complexities of the era is Martin Dillon's 'The Dirty War'. It's very easy to stray into simplifications, generalisations, and propaganda. And yet the reality is complex, nuanced, and morbid.
Brief historical overview...
The seeds of conflict were sown with the Plantation of Ireland. By the mid 1500s England was newly Protestant and paranoid about the risk of an invasion by Spanish Catholics via Catholic Ireland (See the Spanish Armada). To counter this threat Protestant communities were created in Ireland; run by English lords, manned by Scottish labourers... often at the expense of the local Irish. This led to the establishment of the Protestant Ascendancy, which was the formalisation of minority rule by Protestants in Ireland. Thereafter Irish Catholics were excluded and discriminated against.
Come 1916 the Easter Rising occured in Dublin, which led to the Irish War of Independence in 1919. The War of Independence concluded in 1922 with the birth of the Irish Free State and Northern Ireland. This triggered the Irish Civil War.
In the 1950s Northern Ireland was one of the least violent parts of the UK. The problem was that Catholics were still discriminated against, most obviously that they were excluded from many jobs. A notable example was that a young Martin McGuinness applied for a job at his local mechanics, only to be told that he couldn't work there because he was Catholic. He later joined the PIRA and became one of Sinn Fein's most senior politicians.
The Troubles began in 1969 when sectarian rioting escalated into increasingly bloody cycles of violence. The unwillingness of local government to make any meaningful progress on the question of civil rights, along with the massacre of peaceful protesters on Bloody Sunday polarised opinion and increased sectarian paranoia.
This was made worse by the establishment of armed groups whose alleged purpose was the defence of their communities; most notably the Provisional Irish Republican Army and Irish National Liberation Army on the republican side, and the Ulster Defence Association and Ulster Volunteer Force on the loyalist.
The PIRA attempted to wage an insurgent war during 1971-72, but this failed to remove British forces from Ulster militarily, and consequently led to a longer terrorist phase of IRA activity. Many were guilty of unlawful abuse and the murder of innocent victims, from republican and loyalist terrorists to the British army, police, and intelligence organisations.
The Troubles ended in 1998 with the Good Friday Agreement. This agreement was negotiated between the British Government, Irish Government, and all major Northern Irish politicians and terrorists. The PIRA/Sinn Fein was brought to the table by Bill Clinton, while Mo Mowlam managed to drag Loyalists to the negotiation.
The agreement normalised relations, established civil rights, released political prisoners, and removed British soldiers from the border. This was the first time the people of Northern Ireland would agree on something. 71% voted in favour of the agreement in the 1998 referendum.
During The Troubles the border was essentially bandit country, roamed by IRA Snipers (most infamously in South Armagh) and smugglers. The British army was deployed in an attempt to secure the border, but they found this impossible. If it wasn't for the road signs you'd be forgiven for missing it. It doesn't exist as a physical entity.
The prospect of creating a hard border is entertained only by die hard Unionists who would do anything to make Northern Ireland more divided from Ireland, and English politicians who know absolutely nothing about Northern Ireland (Boris Johnson, Jacob Rees-Mogg).
Northern Irish politics is bitterly divided and utterly dysfunctional. Local government has earned a world record for the longest time somewhere has not had a functioning government. They cannot even agree to sit down in the same room with each other, never mind something as complex as discussing Brexit.
Local politics is dominated by the two largest political parties: the Democratic Unionist Party and Sinn Fein, representing [British] Unionists and [Irish] Nationalists respectively. They hate each other and represent incompatible beliefs: That Northern Ireland is an inalienable part of the UK... or that Northern Ireland should become part of a United Ireland.
The population is split almost evenly between Catholics and Protestants, and politics is extremely sectarian. Protestants do not vote for Nationalists, and Catholics do not vote for Unionists. There is a non-sectarian middle, but they are politically insignificant. Because voting is tribal it is very unlikely that an election or referendum will change anything.
There's a few reasons that Scotland's independence debate is peaceful. The most important however is that the United Kingdom was forged from a union between the kingdom of England and the kingdom of Scotland. This has meant that Scots have always been an influential part of the United Kingdom. Which contrasts sharply with the plight of Irish Catholics under English and then British rule.