Either the Note 3 or the G2. I didn't like the UI on my Galaxy S4 and it was pretty Maggy so that left a sour taste in my mouth. That (and the ridiculously huge display) are what made me decide on getting the G2. Another reason, although this may not apply to you, is because of the developer support it has and will be receiving in the next couple of weeks.
There are already stock Android ROM's for the G2 (even on Verizon, crazy right?!) and when the Nexus 5 comes out the ROM's will just get better. I can't say the same for the Note 3. However, if none of this applies to you then it comes down to just a few things: display size, phone size, and UI. Personally, I think LG hit the sweet spot with the G2. The Note 3 is just too big and has an inferior display (IPS LCD > AMOLED). About the UI, LG has done a surprisingly good job. Yes it may look ugly, but so is Touchwiz (they look nearly identical). The difference is that LG's UI is far more customizable and IMO, less bloated/laggy. I can't speak to how the Note 3 performs with Touchwiz, but I'd guess it runs it without any lag.
So, in the end, they're pretty similar. I think you should go buy the G2 (or Note 3) and test it out for a week or two. if you don't like whichever device you bought just return it and try the other. Let's us know what device you choose!
P.S. Excuse any typos, this was typed entirely on my G2.