The most notable add-on is the Push option, allowing for Gmail messages and Calendar reminders to pop up on the home screen automatically. It will definitely deplete the battery faster, but if it is charged, it should keep you updated and on time more often. Note that you'll have to adjust settings both on the iPhone and in your Google account to make this happen.
Another improvement is speeding up the search process, as results will continuously appear while typing, saving you a bit of time or maybe helping your memory if you can't remember the term entirely.
The Google Mobile app is available in the iTunes Store for free.