If one sees a conflict and thinks that there's very little to fight about - they are not looking close enough. Usually, underlying reasons are more complex than what is seen at a glance.
And the roots of the Troubles run deep indeed. The Plantations of Ireland were the British efforts to reinforce their control of newly-acquired land by giving out land to British Protestant settlers. First attempts were made as early as 1550s, but these were generally unsuccessful, and the first truly succesful plantation was the Plantation of Ulster in 1609, during the rule of king James I. This created a sizable population of English-speaking Protestant population in Northern Ireland that was loyal to British crown. This was seen as a introduction of "civilization", as opposed to "barbaric" Irish.
These efforts obviously led to conflict with native population, but rebellions were routinely suppressed, and anglicisation of Ireland continued - for example, Irish Gaelic language stopped being the majority language during the XIX century. The Great Irish Famine also contributed, since it hit the poorer regions of the country harder than the rest. By 1911, there were only 500 thousand native speakers of Irish Gaelic in Ireland, out of 4.3 million people in the census (1). This, by the way, is the reason for the IRA songs being in English - by the time they were written, most of the people who would sing them spoke English as their main language.
This anglophone protestant immigration created the basis for the Irish unionist movement. While the Irish majority wanted Ireland to govern itself (Irish Home Rule), Ulster minority, which identified with Britain, were quite content to be the part of the United Kingdom. This led to the crisis in 1911, when the Third Home Rule bill was introduced to the House of Commons of the UK Parliament. Prevoius bill was passed in the Commons, but then vetoed in the House of Lords. But in 1910 House of Lords' unlimited veto power was replaced by a two-year postponement, so the bill would become law in 1914. As a response, Unionists formed Ulster Volunteers - paramilitary organisation which aimed to oppose Irish government by force. Tensions were rising, both sides were accumulating weapons, and it seemed that civil war was imminent. After the Curragh incident, where British officers threatened to resign as a response to planning military action against Unionist paramilitaries, the Partition was proposed, which would exclude Ulster from Irish self-government.
On the other hand, Irish nationalists, too, formed a radical faction. As opposed to Home Rule proponents, who intended to form a new dominion of the Birtish Empire, Republicans wanted an independent country - even if they had to use force to get it; but most importantly, they wanted all of Ireland, not just the part where republicans were supported by the majority of population. Home Rule support waned, as British seemed to support the Unionists, which resulted in Republicans taking power, and ultimately - to creation of modern Republic of Ireland.
These are the sides of this conflict. Not some amorphous, generalised average "British" and "Irish", who, as you noted, are not that different - radical Irish nationalists, who want unified and independent Ireland, and just as radical Irish Unionists/Loyalists, who want to keep the Irish Republic out of North Ireland. They might drink the same beer and they might speak the same language, but their defining interests are irreconcilable. Both parties engage in terrorist attacks, and each attack by one party spurs the other to retaliation, and the violence divides the sides further. As any civil war, the conflict self-perpetuates, until both sides manage to find a compromise. Right now, the conflict seems to subside, but its underlying reason still exists; and as the beginning of the Troubles showed, it only takes a handful of disgruntled radicals to revitalise the conflict.
That said, IRA did have contacts with the Soviets, and at least one weapons shipment was organised by the KGB during the Troubles; but saying that the whole conflict was generated by Soviets is a definite stretch. IRA and other radical Irish republicans in general weren't very picky with where their support came from, and collaborated, for example, with German Empire during WW1 and Nazi Germany during WW2.