From personal experience with iPhones, the Galaxy S3 and the Galaxy Note 2 (which is basically a big S3), here's my input:
iPhone 5: Large user base that enhances many of its features. More user-friendly to those who are not so great with computers. The app store, while containing less apps overall has a better quality selection of apps. Plenty of accessories available for this phone. Specifications wise, the only way it loses is in 2d graphics. If you break this phone out of warranty, it's much easier and often cheaper to fix.
Galaxy S3: Better 2d graphics, which is used in all applications. Despite having lower specs, it has better performance because it utilizes its ability better. The same apps load and run much faster. If you are good with computers this is the better choice for you because of the increased freedom of the android system. Customization is also easier on the Galaxy S3. Plus it's cheaper than the iPhone. You can also expand the storage with microSIM cards. More built-in browser support for apps. Plugs into almost anything with a microusb cable.
Galaxy Note 2: This is the phone I have. It's by far better overall with specs and performance. It's huge though, pretty much a mini tablet. It'll surprisingly still likely fit in your pocket (unless you're a girl wearing fashion jeans or a dude in skinny jeans). It has all the features of the S3 and more. The S Pen makes use easier. Because of its processing power though the battery life isn't as long but it's still pretty great and I only have to charge it at night despite how much I use it for tasks such as remote desktop connection and 3D gaming. Most carriers offer this phone (assuming they offer it) for less than the iPhone 5.
iPhone: I want something easy to use that won't confuse me and helps connect me to my friends who also use the iPhone.
Galaxy S3: I want top of the line phone that I can make to work my way.
Galaxy Note 2: Look at my huge beast. This comes pretty close to replacing my laptop completely. I'm a huge techie and plan to do techie things. I want to be a mega gamer, I want to compile and render things on my phone. Or: I just am a complete show-off and this phone is currently the best on the market so I'm going to get it.
I might sound like I'm screaming at you to get one of the Galaxy series, but if you have lots of friends you regularly chat with that own iPhones or are not someone who automatically knows how to use a new device, I do recommend the iPhone.