I think the major deciding factor would be screen and S-pen. If you prefer largest viewing area without soft buttons and feel S-pen would be handy, then it's Note 3. But if you have no use/need for S-pen, don't mind slightly smaller screen, I think G2 has some edge like OIS camera, knock on and good visibility under sun light.
G2's UI is highly customizable and there are a lot of useful settings you can change like soft buttons, setting menu layout, notification panel. Some reviewers criticized busy cluttered notification panel. But you can remove Qslide tray to free up more space for notifications. Also verizon version has more notification area than AT&T, international version.
You can't go too wrong with either and it probably comes down to which screen you prefer and use/need for S-pen.