Whichever side of the Apple vs. Android side of the fence you are on, these are probably the two best phones on the market right now (They are certainly the top selling smartphones). So, hem and haw all you want fanbois, buy either of these bad boys and you really can't go wrong. You will have a fantastic smartphone that will serve all or most of your needs.
It really comes down to some very fundamental questions you need to ask yourself if you are still on the fence.
Do you want a big screen at true 720p? Do you want NFC? Do you like the open nature of the Android OS? Do you want universal Micro USB instead of proprietary connectors? Then the Galaxy S3 may be the right phone for you.
Maybe you are a big Apple fan, and are plugged into their eco-system, then the iPhone is going to be a great choice. Like tons of mature apps? The iPhone is going to be superior here no question. And no one can argue about the phenomenal nature of Apple's build and design, it really is a solid high end device. Do you want your OS to be more locked down and simplified, with less options to mess with? Go iPhone.
It's really not a 'which is better' argument, but which fits into what you are looking for, and fits into your needs and desires for a great smartphone. Choice is what it's all about.
Also, take these types of reviews for what they are: General overviews of basic features and functionality of both phones. Don't think that because a reviewer presses buttons at relatively the same time that means one is actually faster than the other. (The one with the bigger screen will take longer because guess what, it has a bigger screen) Look at the specs, and only use these types of reviews as some basic starting research. Do your own in addition to everything you read to get a true picture of what will work for you.
I bought my Father an iPhone 5 for Christmas, and my wife a Galaxy S3, and I have no doubt both of them will love their phones for all that they have to offer.
This video is from pocketnowvideo.