First, I should correct you that both the US and Europe use the International Labour Organization's definition for unemployment: "those who are currently not working but are willing and able to work for pay, currently available to work, and have actively searched for work." Europe implements this as people who are not working, have looked for work within the past 4 weeks, and are available to start work within the next 2 weeks. The US methodology differs slightly, but not in a statistically significant way stating that workers must not be employed, they must have looked for work sometime in the previous 4-week period, and they must be available work work. The last part differs, but since these are surveys, I don't think anyone being asked "are you available for work?" would interpret that to mean at the precise moment. So, the US doesn't define "in the next two weeks" and so some people might interpret it to be a longer period and some might interpret it to be a shorter period, but I don't think you can say that American numbers don't count people that European numbers do count.

So, European and American unemployment rates can be compared with decent accuracy. The fact is that governments want accurate data. Inaccurate data just makes it harder to create policy that helps those governed.

One of the nice things about the American unemployment numbers is that they're more than a gross rate. The US also publishes numbers by certain groups. That allows me to say that the unemployment rate for men is 22% higher for men than for women or that teenagers have been the hardest hit group with unemployment at 20.8% for them. Likewise, there are interesting (if not meaningful) differences in the unemployment rate changes between groups. The African American unemployment rate is up about 30% YoY while the Caucasian unemployment rate is up a whopping 64% YoY.

Full data: http://www.bls.gov/news.release/pdf/empsit.pdf

So, why is American unemployment a bigger issue? You already mentioned it: we've got less good stuff coming from the government when we don't have a job. Healthcare especially is a problem. Beyond that, the amount of time that unemployment benefits last, the amount of each check, etc. just aren't as good. Unemployment is more of a hardship in America than in Europe. Europe has seen very hard times and it created a system that would allow people to get through many years of tough times. America has been able to limit tough times to be much shorter periods and, generally speaking, just has a "get off your butt" attitude.

Partly, this is because America has allowed for drastic economic changes in a way the Europe hasn't historically. The United States is more ready to see old industries die and to start new ones in their place. And so hard times do come (it's inevitable that you can't make something perfect), but they tend not to last as long.

Before I go further, I want to say that much of this is historical. The Europe of today isn't the Europe of even 20 years ago. That said, one way of thinking about this is code refactoring. It takes time, it can be painful as things that worked stopped working as you fixed it. However, arguing that something is working and just trying to (indefinitely) prevent any non-work in order to realign things for a better future is worse.

It's my personal belief that governments can help smooth over the "refactoring" by encouraging new industries to locate near industries that are closing down. Note, this can't be a "we'll just lower our taxes" scheme. There's no point in lowering taxes on a dying industry or throwing long-term government support behind it simply because failure of that industry means unemployment. You need to be smart and find industries that are likely to be growth industries that will progressively demand more labor as time goes forward. Keeping people employed is great, but encouraging industry that increases employment in the long-term is sooooooooo much better. And government can incentivisze new industries. It's hard and often people hate it because they see it as discriminatory against industries not getting the cool incentives, but some industries do benefit communities more than others and some industries are growth and some aren't.

In closing, I'll just say this: in the face of an economic crisis, it might be a good idea to repeal the employer part of the payroll taxes (note the might qualifier since I haven't studied this). It would lower the cost to employ someone by, what, 9%? As the cost of labor goes down, employers want more of it. And when the economy gets good again, it could be reinstated or not and if not would really just be passed to workers as salary increases (just like the employer part of the payroll taxes makes them pay you less each year because they need to pay the government for your employment).