I think it's difficult to compare the two situations because they're so different. You could argue that the Irish have an ongoing grievance that hasn't been resolved though.

One of the major factors that certainly accounts for some of this is simply the great differences in time since independence has been achieved. The USA has been an independent country since 1776, whilst the Irish War for Independence only occurred from 1919 to 1921 and as /u/The_Captainn has pointed out, many Irishmen would view the outcome as unsatisfactory to this day.

The period following American independence in fact saw a similarly turbulent relationship between Britain and her former colony, although for different reasons, as seen through the War of 1812. Britain even considered aiding the South during the Civil War and it is only really in the 20th century that we begin to see a stronger alliance forming, born out of shared interests and mutual enemies.

A major reason is that America sided with the Allies, led by Britain and France, in both World War I and II.

If you look at similar time periods since independence, three are more similarities. Anglo-Irish relations were dominated by issues of sovereignty until WWII. Partition was the main issue in the 1950s but relations were good on a practical level. Although the outbreak of the Troubles set relations back badly, by 1973 the two governments had cooperated over the Sunningdale agreement. In 1985 they signed the Anglo-Irish agreement to formally cooperate in Northern Ireland. By the time of the peace process in 1994 relations were excellent.

By comparison, in 1860 the British were far from committed to the USA.

The thing to remember is that the Troubles were not a dispute between states. While it did lead to disagreement, Irish and British interests were not radically different.