Iphone 5 is one of the best smartphones. Droid Razr Maxx is a little to plasticy and not very high quality. the iphone has usually out done everyone when it comes to quality and software. i personally would suggest the Iphone 5. i have many friends who are VERY pleased with their new iphones. i have an older iphone and trust me, its worth it!
I had the Droid Razr Maxx (the older one that isn't Droid Razr Maxx HD), and my partner has the iPhone 4S (which isn't very different from the iPhone 5).
The Maxx has significantly longer battery life and a bigger screen than any iPhone. They of course also look different; personally I like the look of both of them (and both are solidly built).
Otherwise they're more or less the same in performance, apps, features - except for the operating system, where it comes down to the whole "iPhone vs. Android" argument that you can always see here on Yahoo! Answers... basically my opinion on it is this:
If you already have an iPod or iPad, get an iPhone. End of story. There will be nearly no learning curve and it works with iTunes.
Otherwise it depends on how much of a techie you are. Androids are better for "power users" who like to really customize their phones, iPhones are better for people who just like to keep things simple, or who want to make sure that they are always doing what's popular/fashionable.