I think that, in the US, counties and their boundaries are of most interest to those running from the law ~ once you cross the county line, the cops are powerless to continue pursuit (unless they are state or federal authorities, of course).

Well, that's perhaps an exagerration; it really depends upon where you live. I would think that in the more urban areas, you're much more conscious of your city/town/municipality and your state than of your county, but out in more rural areas, where the towns are so much smaller, you're more likely to identify with your county. (Especially in areas where the County Fair is an important annual event.)

I think counties are more important in Ireland because they are the principal politcal/geographical subdivisions of the country, analagous to the states of the US. I suppose someone could argue against this point of view by citing the four provinces, but I don't think that applies in any real-life sense.

I live in Louisiana, where the political units equivalent to counties are called "parishes." Strictly semantics ~ our (non-eccelsiastical) parishes are "counties." I don't believe the story related above, about an uninhabited Louisiana parish whose seat of government is outside its boundaries. However, we do have one very sparely-populated parish down in the swamps which consists of two non-contiguous areas ~ not sure of its name: St. Landy? St. Martin? I have seen it on maps. The parish government there, of course, would be in one of the two separate areas and not the other.

Counties in the US typically encompass multiple municipalities, but most of our larger cities are counties unto themselves: the boundaries of Orleans Parish, for example, are identical to those of the City of New Orleans. Same thing for The City and County of San Francisco.

New York City is unique in that a single city encompasses five counties of the State of New York: Manhatten is called New York County, I believe; Brooklyn is Kings County, Queens is Queens, Staten Island is Richmond County, and The Bronx is Bronx County.