Stanford professor Robert I. Sutton ably describes the intersection of the Australian and American usage in his book The No Asshole Rule as follows:
- After encountering the person, do people feel oppressed, humiliated or otherwise worse about themselves?
- Does the person target people who are less powerful than he?
Americans seem to have a slightly broader definition of "asshole" that includes people that we Australians would term "fuckwits" or "dickheads." A person can be a "fuckwit" without being an "arsehole" if they show themselves to be generally ignorant about the way the world works in a way that invites derision, but does not do so in a way that intentionally harms or attempts to harm others.
In an effort to further elucidate the subtle differences, I have helpfully prepared the following chart:
Where 'A's are the unintentionally inept, 'B's are actual, literal, anatomical arseholes and 'C's are those that fit Prof. Sutton's criteria (who also just so happen to be fuckwits).
Let me close with a few examples of American usage of "asshole" that would not be acceptable in standard Strine.
Let me know if you have further questions or counterexamples.