I also bought an Audiobook through the iTunes Store on using my iPhone 5S 64GB, iOS 7.0.3, and although it downloaded, it does NOT show up in my list of Audiobooks with the others. I CAN, however, find it using the iPhone's Spotlight Search from the Home screen, and listen to it.
I killed all the apps, rebooted the iPhone to no avail. I connected my iPhone to my Mac, and in iTunes on the Sidebar, I open the triangle to see the contents of my iPhone. The Audiobook shows up in the Books list on the phone. I was able to sync the Audiobook from the iPhone to the Mac. I am not, however, able to sync the Audiobook back to my iPhone, except that the original audiobook remains somewhere hidden, only accessible using iPhone Spotlight.