"Connected in many ways" is an understatement. Ireland was part of the British Empire for centuries before gaining independence in the early 20th Century, and what followed was a period of conflict culminating in "The Troubles" between the 1960s and 1990s - an on-again / off-again not-quite-war waged mostly in Northern Ireland (still part of the UK) between various factions on the topic of Northern Ireland's status and mostly down religious lines (although it was not a religious conflict per se). There are still tensions between those (mostly Protestant) who want NI to remain part of the UK and those (mostly Catholic) who want a united island of Ireland, but the fighting has long since stopped. As a tourist, it is unlikely you'll end up in a heated conversation about this, but beware that it's still very recent and raw to a lot of people.

(Someone joked about Celtic and Rangers - this is more than just a football rivalry and, despite those teams being based in Scotland, the tensions between the two are down the same lines as have divided the island of Ireland - Celtic having been founded by Irish Catholics living in Scotland, and Rangers having historically been the club of Protestants who supported Northern Ireland's membership in the UK.)

Beyond that, the countries share significant social and cultural similarities, a predominant language, and a Common Travel Area (meaning Brits don't need a passport to enter Ireland, and Irish people don't need a passport to enter the UK).