If you're not at least a little jealous of an Android or Windows Phone home screen, then you haven't checked one out recently enough. Apple obviously needn't (and wouldn't) emulate every feature its competitors offer in this department, but culling a few of the best features and implementing them on iOS would substantially improve the experience for basic and power users alike.
An Android home screen can include widgets like a weather forecast, quick access to settings like Bluetooth and Wi-Fi, highlights from your calendar, tweets, an omnipresent search box, and plenty more. And Windows Phone dumps the familiar icon approach almost entirely, in favor of tiles that can reflect far more data than the iOS home screen: Live Tiles on your home screen don't just provide quick access to your email--they can display the latest messages in your inbox. The calendar tile shows your next appointment. And Live Apps can reflect their latest content (your timeline, news headlines, and so on) on your lock screen, without your needing to update them manually.
Organizing apps on iOS is still painful, whether you try to do it on your device or from iTunes on your Mac. Windows Phone uses the familiar "tap and hold" to organize approach, but the organization process still feels considerably simpler because Windows Phone uses a single, vertically-scrolling screen for organization. While that would seem to get unwieldy if you have hundreds of apps, it works for Windows Phone because that platform treats its home screen more like Mac OS X treats the Dock: It holds your favorite stuff, and you dig deeper (via search or an alphabetic list of all your apps) when you want to get to the stuff you use less frequently.
While these approaches will strike longtime iOS devotees as foreign, that doesn't make them inferior. I suspect users with more than a couple screenfuls of apps, or with more than a couple of indistinguishable folder icons, would welcome such innovations from other touch-based operating systems.
Sharing data and documents between apps in iOS still stinks. Apps that save documents to iCloud, for example, can't directly share those documents with other apps; the best you can do is open a copy of your document by starting from the originating app and using it to send the document to a second app.
It's unintuitive, clunky, and frustrating.
These same data- and file-sharing problems afflict apps that use local document storage instead of relying on iCloud: You can send only copies of documents between apps--a serious hassle.