Well, one thing is certain, even during the bottom of the recent recession the unemployment picture in the US was better than anywhere in Europe.
One thing that Europe is probably "better at" is that a higher percentage of stable jobs- people on average stay at one place of employment longer- socialism gets this result. And this gives a false sense of job security. If you looked at only govt employees in both US and Europe you would see this stablity- govt employees commonly work all their lives in govt jobs.
In US you have a lot people leaving their jobs and getting new jobs- and it's likely that this trend will increase. This churning of jobs gives the individual perception and fear of unemployment. [Of course fear for some is fun for others- few people dream about being stuck in one job for the rest of their lives.]

Couple of points. We are in the Information Age. Knowledge means power. Changing jobs generally requires someone to gain more knowledge. Hence the America workforce on average should be gaining more knowledge than European workforce.
What you have in America job market is more diversity than Europe. And like in ecology, diversity gives a survival advantage. There might be more fear about unemployment in America, but this is more to do with change than actual economic well being.
Finally, to whose advantage is it to have a churning job market- I think it benefits the employer and the employee- but the main advantage goes to the employee. And it makes it more likely for employee to become the employer. Or another way to say this is that maintaining a workforce is an advantage to the employer- it's a benefit which they provide incentives to maintain, but generally speaking these incentives are less in value, than compared to the employer quitting his job and seeking another- fear was on the side of the employer- fear of uncertainty. Therefore an employer with fear of change on his side can offer less to keep an employee.