The Galaxy Note 3 has a 2.3GHz quad-core Snapdragon 800 processor and it shows, especially with 3GB of RAM. Because of this, you can launch several applications without noticing any lag whatsoever. You will definitely notice the power of the Note 3 when browsing the Internet; there is no lag when zooming in and out. The processor will play just about any video you throw at it without any stuttering whatsoever. You may have video sound issues, but can fix that by downloading the appropriate codec.
The iPhone 5s 64-bit A7 processor works well, but the "64-bit" part looks much better on paper since there are barely any apps that can take advantage of the processor right now. The iPhone 5s will also need more RAM (the 1GB RAM isn't as bad as it sounds though) to take advantage of the processor. Still, the iPhone 5s operates super fast -- just as fast as the Galaxy Note 3. There is definitely a difference in performance between the iPhone 5s and the iPhone 5.
This category results in a tie; both phones offer the incredible performance that nobody would have predicted just two years ago.
The Galaxy Note 3 run's Android's latest operating system, Jelly Bean (Android 4.3). On top of it is Samsung's TouchWiz operating system. Both Android and TouchWiz have come a long way over the past two years and even though the Galaxy Note 3 will confuse new smartphone users, veteran geeks will easily adjust to all of the capabilities.
On the other hand, iOS 7 on the iPhone 5s is for beginners and experienced smartphone users. Apple made a major jump in simplicity from iOS 6 to iOS 7, even though some may not realize it until using iOS 7 for at least an hour. Running the iPhone 5S with iOS 7 is the most fluid smartphone experiences one can have.
The winner: The iPhone 5s wins this round, although not by much. Android's operating system on the Galaxy Note 3 has greatly improved from previous Note devices.
The Galaxy Note 3 is a multimedia marvel and watching videos on the 5.7-inch screen is surreal. Unlike the iPhone, you can play just about any video format with the Galaxy Note 3. Listening to music doesn't give you the full richness you will experience on the iPhone 5s, but it is good enough so you won't have to buy an extra mp3 player.
We especially liked that the Galaxy Note 3 allows you to shoot 4K videos. Although 4K devices aren't mainstream yet, we played back one of our 4K videos on a 4K set at Best Buy and the results were quite amazing. Watching the video was just like being there.
The iPhone 5s is a great multimedia machine that his hampered buy its too-small 4-inch screen. Still, it is the best smartphone you can buy to listen to music; the sound is even richer than it was on the iPhone 5. The 5s takes amazing pictures for a smartphone, but the video quality can't compare to that of the Galaxy Note 3.
The Galaxy Note 3 wins this round. However, both devices are excellent multimedia devices. We suspect Apple may win or tie this round when they release a phone with a bigger screen.