short answer: The keyboard and lack of push email.
long answer: While on the road recently, I was forced to use the email functions in the iphone more than I had in the past. The vertical mail keyboard is too narrow for my big hands and often the predictive software slows down the process of writing a quick email. The lack of feedback from the screen hurts here. I also get really annoyed by the constant delays in response do to client server communications. The “typewriter” effect seems to jam up like the very machines they are emulating!
solution: as a temporary fix, they need to let the keyboard go horizontal in the mail app. This will lessen mistakes and speed up the use of the keyboard and lower reliance on the predictive software.
The long term fix is to use a slider with a real keyboard like the latest sony device or some of the LG units.
Knowing how Ive and Jobs solve problems, my thought is that the next gen of the hardware may have some sort of integrated (well designed!) keyboard that makes interacting with text more intuitive. When the speeds promised in the 3G handsets reaches ubiquity a solid input interface will be critical to the iPhones market penetration. Apple will have to eat humble pie (or come up with a phenomenal voice entry solution!)
Most of you that have had the device for a while now and are heavy users have probably found these same issues. Let us know what you think.