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OK, to begin with, Australia still bans things. This is something the USA grew out of decades ago:BULL and SHIT. In Australia, there is no routine pixellation of middle fingers, likewise pixellating the mouth of someone swearing.
It's an utter fabrication that there's no censorship on things in the US: It is a violation of federal law to air obscene programming at any time. It is also a violation of federal law to air indecent programming or profane language during certain hours.", from http://www.fcc.gov/guides/obscenity-indecency-and-profanity
And I've already mentioned the Press Freedom Index, which indicates the Australian press is freer (scum that they are) than the US press. There's a difference between idealism and how the place is actually run. Hell, the US even introduced the anglosphere to the concept of "Free Speech Zones".
Hell, even movie ratings are a form of censorship, and the US has those. Those movie ratings cause movies to be modified in order to meet them. The US even introduced the world to "Parental advisory: explicit lyrics" which was a soft form of censorship; some stores would not carry such marked CDs, and some shopping centres would not allow stores within to carry them.
Some US states prevent atheists from holding political office - that's an explicit penalty for exercising freedom of speech.
When I was in San Francisco, it was illegal for me to take a photo of a school. Standing in a public place, it was illegal for me to take a normal point'n'click photo facing one direction but not another. Similarly, both countries ban public nudity except in some specific areas.
Then, of course, there's all the provisions for the support of US IP law and legislating against filesharing that Australia has had to engage in duty to treaty obligations with the US.
Ultimately, when you say "Australia still bans things", you make it sound like a commonplace experience. It really isn't - and banned items are few and far between. Your wikipedia link has little that Australia has banned in the past couple of decades, and includes paragraphs like: Explicit sex scenes have also become more common. Channel Nine's crime drama Underbelly has frequent coarse language and sex (including explicit anal rape) with one episode featuring a "drug-fuelled orgy with prostitutes accompanied by the Spiderbait song "Fucken Awesome"". I am actually interested to hear if you can name more than a handful of banned items from the past couple of decades. By the way, does exporting cryptography count?
The US does have better protections for individual freedom of speech, but the gulf between the two nations really isn't all that large, nor is it all in the US's favour.
-- Which are similar to the insane laws against denying the Holocaust or even using certain images in parts of Europe:
One thing that we here in the Anglosphere need to remember is that in WW2, we were unequivocably on the side of moral good. We were fighting the clearest moral evil humanity has known. All of the Anglosphere, undivided. For the European continent, the experience was much more complex and painful, and even those countries that fought against the Axis had plenty of collaborators. Idealism is nice... until it's tested, then pragmatism needs to have a voice.
In any case, you're talking about a very specific incident in history, which in Europe has a very real knock-on effect of lending support to neo-nazism. You can go hog-wild about denying the efficacy of Bismarck's reforms. Question Kaiser Wilhelm all you want. But the pragmatism of legislating against holocaust denial or sporting the swastika is basically the same as anti-defamation legislation: it's saying 'don't diminish this'.
Think about it this way: those laws are very specific in terms of what they ban. So you can't show the swastika in Germany. It doesn't mean that they're going to 'slippery-slope' their way to banning other images. There seems to be a belief in America that Germans want to obscure the past - they don't. From a friend in Germany, they're acutely ashamed of that past, and these laws help to avoid obscuring that past and lessening the impact of the resulting lessons. The laws you speak of here aren't things that hit people in their everyday lives, they're specific, targeted laws. The laws would not be around in an ideal world, but in a practical world, there is a tangible reason for them. As laws go, they're not 'insane'.
I'm not sure how the side-effects you see of these laws is any better or worse than the routine emotional pain that folks like the Westboro Baptist Church deal out under freedom of speech laws. Not showing a swastika in a manner other than historical education, compared against having a lunatic scream at me while I'm burying a loved one? Which one actually has the more social or psychological harm there?
-- What's worse, they do nothing to solve any of the real problems modern Europe has with anti-Roma and anti-Muslim racism.
US free speech also does nothing to solve any of the real problems modern America has with anti-Mexican and anti-Muslim racism. The holocaust denial laws are not there to solve racism. They're there to stem violent nationalism, the European forms of which don't exist in the US.