The iPhone does video better than the Note 3 and blows it away in low light. The two devices aren't even on the same category, imo. Low light performance is one area where Samsung has failed consistently, and they only do slow motion video at 480p resolution which is worthless for anything you'd need to use that for.
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There are three areas where iPhone falls behind in video capture :
1.Mono Audio track and Poor Microphone Quality (Poor sound quality)
2.Lack of 4K recording
3.Very poor user control (All settings are locked down pretty much , If you carry a 16GB/32GB iPhone and want to take lot of 1080p video , its questionable situation , There is no option to set video to 720p for capturing more . locked down to 1080p)
Galaxy Note 3/S4 falls behind here :
1.OIS is terrible
2.Low light is questionable and flash is unnatural
3.Launch time more (but provides lot of settings for users though , its a trade off)
Overall , Both Cameras have cons , it comes down to users to pick their poison.