[Galaxy Note 3 Q&A, Help & Troubleshooting] BarryH_GEG:
It depends on what you're looking for. If you have a G2 I'm guessing you like overlays and the features they provide. The N3 has a ton of s/w features aimed at productivity and creation users centered on the S Pen. If you're one of them there is no phone that compares to the Note-series. Both phones have "gee whiz" features with things like motion control, Awake Stay, and the like. So if you like "stuff" the N3 has lots of it. If you're going to strip both phones of their features and use an AOSP-ish ROM than pick the device that makes you happiest. The N3 has a removable battery and expandable storage so keep that in mind as a possible benefit.

Also keep in mind that two philosophical issues affect the N3. To prevent gray market sales Samsung's introduced Region Lock which requires a device be initialized using a SIM local to its selling region before you can use it globally with different SIMs. There have been some issues where people have done that and run in to problems. The other is all Samsung devices now come with KNOX security s/w preloaded. It's benign when not used but Samsung moved the flash counter to it so when non-standard s/w is run on a KNOX-equipped device a flag is tripped that allows Samsung to know that and potentially withhold warranty support. There are long threads on both of those items if you want to learn more.

The speaker on the N3 is louder than the G2's but a bit thin and lacking in detail. If you're a big external speaker user you might want to test it in person to make sure it meets your needs.

Both the G2 and N3 are made of plastic so there's always the chance that long-term durability could be affected by handling. It's too early to tell.

It's apples vs. oranges. The G2 has a great LCD display and the N3 has a great AMOLED display. Which do you prefer? They both have strengths and weaknesses. Here's a review of the N3's display.

http://www.displaymate.com/Galaxy_Note3_ShootOut_1.htm

As sturdy as a plastic phone can be. Mine's certainly sturdy with no squeaks, rattles, or unnecessary flex. The back cover's paper thin plastic to accommodate being peeled off for battery, SIM, and SD card access. Some people hate that design others are fine with it.

The N3's better based on GSMArena's objective testing.

Audio quality

All three smartphones delivered almost spotlessly clean audio output so picking a winner here was quite hard. Still, the Samsung Galaxy Note 3 managed to take the victory with its louder output, and slightly lower stereo crosstalk when headphones were plugged in. The LG G2 and the Sony Xperia Z1 are basically identical as far as their audio output is concerned. They perform excellently both with and without headphones but are let down by their uninspiring volume levels.​

http://www.gsmarena.com/lg_g2_vs_samsung_galaxy_note_3_vs_sony_xperia_z1-review-1000p7.php

It depends. I have a ton of syncs running and leave all Samsung's motion, air, and audio sensors active. I'm also in a crappy LTE coverage area. I get about 15-20% less battery life than I did on my non-LTE N2. I get about 5 1/2 hours of screen on time and the N3 easily lasts through a heavy use day. It also charges insanely fast so topping up during the day can get you through two days of use easily.

Mine hasn't but I'm not a big gamer. It'll get warm when I'm background downloading while doing something resource intensive in the foreground but it’s never gotten hot.

There's no shortage of opinions here. ;) There are about a dozen people on the forum that post their displeasure with the N3 when the opportunity presents itself. Quite a few are former N2 owners. Weigh your own needs and hot buttons against the positive and negative comments this thread will attract and draw your own conclusions.