I can replace the on screen keyboard with one that has a full five-row keyboard for times when I SSH into a machine. On a similar vein, when I SSH into a machine, I can actually leave the SSH session running in the background while I switch to another app, without fearing that the OS is about to kill my SSH app while I'm looking something up or responding to a text message. I can also leave my IRC client running in the background without it constantly needing to reconnect when I switch back to it.
Intents in Android, especially in combination with the global Share mechanism, allow any app to receive arbitrary data from any other app, meaning apps don't need to know about specific apps or services in order to integrate with them. Clicking on a URL allows you to choose which browser (or set a default) to open the link in, allowing you to use alternate browsers (or alternate email clients, SMS apps, dialers, etc); tapping Share in the browser allows you to send the current URL to any application that can receive a URL, making apps like Instapaper, Pinboard, and 3rd party Twitter clients have the same capabilities as first party applications.
Sideloading apps means I'm not limited to installing programs from the Android Market/Play Store, and can do things like buy apps directly from the Humble Indie Bundle and install them on my own.
Proper background service support, and allowing apps to affect things outside their sandbox, lets me run programs like Locale  that can monitor the phone's status, location, etc, and modify the phone's settings automatically based on a set of conditions that I've pre-arranged. My phone automatically silences itself at night time and while I'm physically at work, turns on my Wifi when I'm at home or work while defaulting it off when I'm out and about, and more.
That's just some of the reasons I like Android better than iOS.