The Union Jack would be illegal.

The world would be vastly different given the different approaches, priorities, and values of an Irish Empire.

Potato would be the most consumed good

Whole wars would be fought over the question of whether blended whiskey is better than single-malt, and Oxford would offer PHDs in "peaty-ness".

Drunker

they would be called Potatoies

not Limeys

Bhíomar go léir ag ithe prátaí don lón bricfeasta agus dinnéar

That’s Scotland.

You're right. All the Single-malt snobs in Ireland would be dead before the Bushmill's/Jameson wars came to their bloody end.

EDIT: NOTE: In the world suggested there was no Great Irish Diaspora, so all of the annoying Irish-Americans are still just Irish.

Well, for one it would have been a Catholic Empire instead of a Protestant Empire. The Inquisition didn't end until the 1800s, and since the US colonies would have been all Catholic then the Inquisition would have continued in North America, and it's possible it may have never ended. The Americas would most likely be primary Catholic as well instead of Prodestant.

If this ended up being the case, it is easy to imagine that at least one more Crusade would have been launched over the past 500 years.

One might question if the US would have seceded from the UK at all if it was founded by Catholics, or if The Holy Roman Catholic Empire would have beaten Napoleon and conquered all of Europe, North America, and South America.