I think it is pretty darn important, but of course, it depends on the app (location services etc.). My wife is really bad about leaving apps open, and every time she complains about battery life, I look and find 15 or 20 open apps. I close them and she doesn't complain again until she leaves another dozen or more running in the background. Personally, once a day, when I'm done using the phone I close everything I don't need running and I have never had a battery issue, even with the 4S.
I close apps because despite people claiming open apps doesn't affect battery, it sure seems like they do.
Anywa, I think iOS is really missing a Kill ALL switch in the notification slide or even a quicker way to shut an app immediately after use because the process of killing apps individually is annoying.
The apps themselves are not actually running in the background. When you switch to another app or go to the home screen the app you were previously running enters a suspended state. This means its is still in memory, but it's not using the processor. It will remain this way until you either reopen it or iOS automatically closes it to free up memory as necessary. Note that the app's icon will remain in the multitasking bar even after the operating system has closed it.
Now while apps do not run, iOS does have a number of services built-in that allow an app to continue certain tasks while in the background. "Closing" the app will discontinue these services.
I remember when app switching was first release Apple said that they weren't using any processor power in the background. Yet, when I went to the Genius Bar at my local Apple Store several months ago to complain about a home button on my iPhone 4 that wasn't working properly, the first thing the Genius did was check to see how many Apps I had open. He closed them all and told me that the issue with my home button was actually lag caused by having too many apps running.
I said that I didn't think apps actually "ran" in the background and he gave me some answer that didn't satisfy me. Regardless, I found it odd that an Apple Genius was suggesting closing apps when the original presentation given by Apple HQ said you didn't have to worry about it.
First thing I do when an app starts to act buggy or become unresponsive is close the specific app or all apps. This solves the problem 99.9% of the time. Cause of this I never had any reason to reboot.