The most significant difference is that in the US unemployment benefits are generally limited to a maximum of 26 weeks, while in Australia you can collect benefits for decades.
The "greater benefits" available in some states (e.g., Massachusetts) are only available to people who have been recently working in a high-income job. Your unemployment benefit in MA = 50% of the bi-weekly pay rate you had when you were working, up to a certain maximum. And, again, you can only draw unemployment for a maximum of 26 weeks.