Having both the Note 3 and the 5s, I much prefer the 5s.
First, I don't have to perform hand gymnastics to use the 5s one handed. My hands are big enough that I can use the Note 3 one handed, but damn my hand gets a workout from it (6'2'' and bear paws for reference).
Second, the stylus on the Note 3 is a gimmick. It only really works for the samsung apps, and unfortunately not very well even with those. Otherwise it becomes a point and tap annoyance.
Third, the "multitasking" sucks on that screen size. I have had multiple crashes, but ignoring that fact, using two app on a small screen does not work well. Can you do it? Yes, you can. Is it a good user experience? No, it is pretty abysmal.
The Note 3 itself is nice. It has good battery life, and the screen is very responsive for an android device. Hands down one of the best screens I have used on an Android device. Unfortunately, Samsung loaded it with crap in my opinion. Way to many gimmicks and finicky features that sometimes work and most of the time cause some sort of issue. If they had this type of screen on a GS4 form factor and size, lose the S-Pen, and clean up their apps they will have a stelar phone. Until then though, the 5s is a superior all around experience.
Obviously this is just my opinion from using both phones for the last four or so months.
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