Bushmills and Jameson are the two Irish whiskeys you are most likely to run across and they are both very good. You'll like either of them if you like your whiskey clean, crisp, and pure. Why? I'm glad you asked. ;-)
Irish whiskey is made from malted barley that has been dried in a "closed" kiln. Scotch, on the other hand, is made from malted barley that has been dried in an "open" kiln. The difference is that the open kiln used for scotch allows the smoke to come in contact with the malted barley. The flavors the smoke imparts to the malted barley are what give Scotch its tantalizing complexity and sometimes, even, the sense that you are drinking something found in a house that burned down. None of that for Irish whiskey though - since the smoke never touches the grain in Ireland, the whiskey has none of the smokey complexity of scotch - instead it comes out clean, crisp and pure. Some people like complexity, some people like crystal clear, clean, and fresh Irish whiskey. Scotch and Irish whiskey are both aged in used barrels so they pick up somewhat less "oak" flavors than bourbon, for example.
While we're on the subject, bourbon is made from at least 50% corn, which has a much higher sugar content AND is aged in barrels that have never been used, which gives it its much sweeter, richer flavor. If you aren't intimately familiar with the three, do an interesting test. Drink either Irish whiskey or scotch all night and then, for your last one, pour yourself a nice Kentucky bourbon. It will be so sweet as to almost be cloying.
But if you are into the Irish whiskey, Bushmills is derned good, and if you want an extra special treat, try their Black Bush. Or go all the way to the top and check out their single malt's.
Another aside, for those who might care about these things, the Jameson distillery is in County Cork (read, Catholic) and the Bushmills distillery is in Northern Ireland (Protestant). While I believe that both brands will go to the end of the earth disavowing any lingering loyalties or affinities, some people see one or the other as traditionally "theirs".