Thoroughly loving the Note 3 after having it for a weekend. Perfect size for my hand, I can still hold it with one hand and reach my thumb anywhere needed, even swiping the notifications down from the top of the screen. Alternately, there's no way my wife could use this thing 1-handed, it's way too big for her. After using my Note 3 for a little bit, she's definitely decided on sticking with the Galaxy line of phones, her S3 fits her perfectly.
The battery is pretty solid, I don't know that it's significantly better than my old S4 though. Seems like through a normal day of usage, either phone was at 30-40% left at the end of the day, with gaming draining it much, much faster. I guess it's all relative, larger battery + larger screen = roughly same amount of usage.