I used the note 3 for my daily phone for 4 months, which is a long time for me. It's a great phone. I thought it slowed down the longer I used it. Touchwiz is getting a little old to me. Been using the g3 for a week now, and it is truly amazing that they fit a 5.5 inch screen in such a small footprint. Just for the size alone and the cleaner ui I would go with the g3.
The Note 3 is $50.00 more than the G3 on contract (Verizon) and almost a year older than the G3.
That being said, they are both excellent phones and it will come down to user preference.
How's the battery life compared to the note 3?
The note 3 has a beast for a battery. I am a pretty heavy user, gps, lots of phone calls, email, text throughout the day. If I don't charge during the day I will be at 25% around 9 in the evening. Which I think is pretty solid. It's not as good as the note, but it's better than I was expecting with the screen.
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depends what you want out of a phone..
I have the N3 and I am going to eventually switch to the G3 once I find a good enough deal on Swappa. I found I don't use the S-Pen at all, and I am disliking touchwiz the more I use it. Also my N3 drops to 3G a lot for no reason (I got a replacement and still did it), so I'm kinda just ready to move away from Samsung and change it up