I don't understand why some people find this so difficult to understand.
Apple's not fighting for people to not call mobile applications "apps".
What they're doing is claiming that "App Store" denotes a particular experience (quality, ease, design, etc) and brand tie -- in people's minds "AppStore" brings to mind Apple and iPhones.
They've been using this name and mark for years now (you know the "avail in the App Store" badges we've grown accustomed to seeing? The ones that tell us immediately it's an app we can get for the iPhone?), and the claim is that if other companies start using "Appstore", or some variation thereof, it will create brand confusion for customers.
Frankly, they're right. It will cause confusion.
Now perhaps some people think this is just too bad, and Apple should rebrand as "Apple App Store" or something of that nature. That's a fine position to have, but to claim that Apple's concerns are unwarranted is kinda ridiculous.