The purpose of Firefox Home is to give you, the Firefox browser user, a way to port a lot of your Firefox "stuff" over to your iPhone. This "stuff" includes your browsing history, your bookmarks, and your open tabs. The first two features are a no-brainer for me. Firefox already has a robust platform for syncing your personal info across browsers. If you have a lot of commonly-visited sites, you can quickly call them up from this app and surf in Safari.

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The open tabs syncing feature is much cooler for me, personally. I frequently leave a whole bunch of tabs open to represent projects I'm working on. Back in the day, open tabs used to eat up a lot of RAM, but most modern browsers have fixed the RAM leaking issues. Anyway, now you can access that digital "to do list" from your iPhone too. On the
Firefox Home page, the Mozilla team demos this function with information about a flight that the user left in an open tab. After accessing Firefox Home, she is able to see that very same tab on her iPhone.
Here is Mozilla's promotional video, for the visual learners out there!
One last feature of Firefox Home is the Awesome Bar. While Safari does have some "awesome bar" functionality (addresses and bookmarks are combined but not search), searching from the Firefox Home app lets you conveniently search with the full power of your Firefox search history and bookmarks behind it. Personally as a Windows user, I don't bother syncing bookmarks onto the iPhone's Safari browser, so the native bookmark feature does little for me.