What you can do in the background is very limited. True, or lets call it unrestricted to be precise, multitasking is not allowed on phones to conserve battery life. If reading about it in dev documentation isn't your thing, listen to this weeks Accidental Tech Podcast. They talk about how limited it is and why it is that way.
Playing audio in the background is the only thing that works that way. Other background tasks are limited to short timeframes at limited intervals to prevent resource hogging by an app.
Yup. The device and OS are totally capable of "unrestricted" (I call it free-for-all) multitasking; it is a derivative of OS X after all. But developers can't really be trusted to respect soft (not enforced) limits on resource usage. So Apple has taken the approach of giving developers specific APIs that Apple themselves create and curate, that respect limited resource usage, but also offer up solutions for most use cases. Its a much better design and provides for a much better and consistent user experience.
Using these APIs you have a few options as a developer. You can do whatever you want in the background for a short period of time (about 10 mins IIRC) after which your app is forced to terminate. If your background work is going to take longer than that, or needs to happen indefinitely, you have the options of using one of Apple's pre-built APIs for things like: GPS navigation, audio, uploads and downloads, background refresh, push, and now also video. You can complete almost any conceivable workflow using these APIs, but you're also forced to respect the resource limits of a mobile device. Android also takes a very similar approach for the most part.
This new split screen multitasking doesn't even break these rules, the only thing that's different is that 2 apps can now have focus at the same time. Once you swipe one off the screen, it goes back to following the same rules as all other background apps.