From what I can tell, Apple's multitasking APIs cover pretty much every multitasking use case that I need, assuming they work as advertised. The main problem compared to Backgrounder, is that Apple's APIs have to be actively supported by the app developers, whereas Backgrounder works with any app without any special support. So unless/until app developers of all the apps I use add multitasking support to their apps, Backgrounder will have an advantage.
One of the main uses I have for Backgrounder is allowing audio apps, like Pandora and Stitcher, to run in the background. I'm sure the developers of those apps will very quickly add multitasking support, since it's such an obvious benefit.
However the other reason I occasionally use Backgrounder is when I'm filling in a dialog box or am in the middle of some other process in the app that I don't want to interrupt by having to restart the app, and then I need to do something else on the phone, like look something up. In theory the Fast App Switching API should be able to handle this, but only if the developer has added support for it. Since some of these apps aren't ones which you would normally think need multitasking, I'm not sure the developers will add support, unless Apple has made it trivially easy to add support for Fast App Switching. So Backgrounder may still beat Apple's APIs for some time.
+1 for the first and third paragraphs.