Android Does more than just SIMPLE Things like your example
Play an audio from a video, in the background.
iOS can do it. Apple disabled it(or maybe it was a bug) in iOS 6, but in iOS 7 you can do it again.
Neither Android, nor Windows Phone allow you to do this little, yet such a handy feature.
Okay, so we have heard your requirement, but to state that Android won't allow such a handy feature is incorrect. As others have pointed out, it is possible through the flexibility that Android offers both in choices of apps but IMO, more importantly, your
CHOICE of devices. While Apple is busy giving you choices of colors the various android manufacturers are giving you real choices weather your a minimalist, want large phones, consider photography key, etc.....
So, can you listen to documentaries, or lectures, or listen audio books? Ambient noises on Pandora, Spotify or Podcasts? Besides, why should I need three different applications, when I can use none and just use the browser.
Your last sentence is ironic considering just about everything you need to do with iOS needs "another app" such as ...... However, let's stay on topic and just state that if you are looking to Android, I would suggest buying the correct phone to do your job.
There's so many choices in Android based phones for a REASON. Some androids do what you're looking to do far better than other models and FAR, FAR better than iOS devices. Please don't make me repeat my posts about multitasking on iOS vs Android.
What a lot of people do is: Open up YouTube in Safari, background it, then launch Chrome and browse the web in it, or do something else, while a documentary or something else is playing in the background. It's very convenient.
What's even better is get yourself a Note 3 and open two windows and truly multitask. Don't want two windows open, then use an app like Viral. I personally prefer two windows. If you really want to though, put the Viral App in the background, Chrome on part of the window and then pop a note taking app up and experience
true multitasking. That's just another in a growing list of things iOS devices won't allow you to do.
(See the last Image Below)
Watch a video and surf or do email, text, whatever.....both live at the same time. You can re-size or move the windows around any way you want in portrait or landscape mode too.
You can even surf the web, watch a video and search YouTube for more time killing fun all while still playing the video as shown in the bottom right corner here. Let me know when you can do 3 things at once on iOS or if you could even see all that on such a small screen like what an iPhone offers.
Take Notes in either handwriting or typed fonts of your choice and size.