What Should a Mobile OS Do?

Let's face it: A mobile Operating System (OS) should do everything a real OS does.  The only difference should be the screen size.  I'm not saying that every phone should be running PhotoShop (Yet), but a mobile OS shouldn't be limited by an app store either.  The only limitations should be the battery life and the processor.  This brings me to my point: What is Apple gaining by running a closed app store?  Shouldn't the apps be treated the same as in a real OS?  Why would you block apps that people WANT on their phone?  If I can put any app I want onto my phone, then it is truly my phone.  Microsoft may have failed when it came to the mobile interface (a scaled down Windows doesn't work),  but they don't force you to use their app store to get an app onto the phone.  What Apple does with their app store is the equivalent of only selling Mac software in the Apple Store and approving only what they want.  That's right, no Microsoft Office for you!  Perhaps the only thing holding Apple back from this is third party plugins like Flash.  Oh, wait.  Maybe not.