At this point, the hardware is sort of whatever. It'd be great if Apple finally offered a front facing camera higher than the pathetic 1.2 MP; it'd be great if Apple finally offered a higher MP count than 8 (sounds like they're going to 12MP); it'd be super great if there was more RAM for those who browse with multiple tabs; wireless charging or quick charge would make things more convenient for people; and etc.
But that's sort of not as important compared to the real "catching up" that Apple needs to do: SOFTWARE.
-Apple Maps is still pretty poor (compared to Google Maps).
-The stock keyboard is mediocre. Cannot keep up with fast thumb tapping, and doesn't offer gesture typing out of the box. Third party keyboards... have all the bugs been hammered out?
-Allow a bit more freedom and customization. Let us choose what toggles we want in Control Center and where. I don't use AirPlay; let me take it out.
-If you're allowing us to set third party keyboards, why not allow us to set third party other things? Google Maps for navigation would be great.
-Give us access to files
-Give us full email attachments.
-Let me share to more things than just Twitter and Facebook. (On Android, not only can you share to anything; it remembers what you share to the most and keeps it near the top).
-Let us access Siri hands-free anytime, instead of only when it's plugged into the car (talk about a half baked feature).
-Let me update software over data, instead of only on WiFi.
-Let me make App Store purchases without forcing me to use a password each time. Let me have the option to turn that safety feature off. I'm an adult. Let adults manage their own phone and expenses like they do other accounts.
-Let's have true multitasking (the RAM would help this).
-Make going back more consistent. I can't stress how much this bothers me when I use my iPhone after using Android. Navigating through iOS is so inconsistent: sometimes the back button is upper left or right, or lower left or lower right, or lower center. Sometimes I can gesture and other times I can't. Utterly frustrating.
-Organize that Settings page. It's a mess. And make Settings more consistent, too. Sometimes Settings are in the app, sometimes they're in the iOS Settings. What's with that?
-Extra little conveniences like double tap to wake and whatnot would be nice.
-And so forth...
There's probably more I can add if I thought about it.
There is still plenty Android can improve on, too, but I feel like iOS has far more room to go than Android. I said this before, Android is approaching desktop-like powerful (I can essentially minimize apps in TouchWiz, or resize the app window which is very desktop-like, and still be able to do things in the background and foreground). Whereas iOS is still a glorified app drawer. The "world's most advanced mobile operating system" just doesn't feel that advance.
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Just to give an example of Android approaching desktop-like capabilities (Galaxy S6):
Yesterday, I was on Chrome on my phone chatting with a customer service rep for an issue. They asked me for the order number, and I didn't have it but I didn't want to risk losing the chat window if I exited out of Chrome (sometimes that happens cause these chat boxes are finicky). What do I do? I minimized Chrome (I think it's called "Pop Out" feature in TouchWiz? Not sure), and in the background, went to my Gallery where I had screen grabbed the order number earlier, saw it in the background, and just went to Chrome in the foreground and typed it out for the CS rep. Boom. I re-enlarged Chrome to full screen and carried on. Didn't skip a beat. Simple, fast, effective, powerful.
(Now that I think about it, split screen would have worked too!)