A managed bean is simply a JavaBean that you've specified to the JSF config file as being automatically constructed and initialized by the JSF framework.
A backing bean is a JavaBean that's bound to a JSF page, usually so it can display and/or input data.
More often than not, backing beans are managed beans, since as long as the framework's willing to set up your beans for you, you might as well let it. For one thing, the management framework will allow you to tie together complex interconnetions between beans without your having to hard-code those connections within the beans themselves. Which simplifies the beans and makes them easier to reuse and to test.
An IDE is no substitute for an Intelligent Developer.