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Help me choose a new phone!
I am considering between LG G2 vs Samsung S4 vs HTC One vs Xperia Z

Which one would you buy and why?

I purchased the Xperia Z1 myself, and I'm really really happy about it. Yet it pretty much depends what you are after:

Size?
Camera?
Battery?
Custom Android?

Size wise, whatever enters in my jeans pocket was more or less fine.
Camera was a main point, and the Z1 as one of the best phone camera.
Battery was another main point, I want as much days possibles out of the battery. Usually I get 5 days without much issues. Keep in mind, it depends what you do with your phone to. If you play the whole day long then don't expect 5 days
Custom android that was less of an issue, and my choice was a more conservative one here. Don't want too much garbage on it.

Note 3..no other phone matters....

I would go with the Galaxy S4 (the Note 3 is basically a larger, slightly beefier S4, but it is also pricier)

I would not say that. The S4 does not have a pen,(which is VERY awesome), the fact that it is bigger makes it not so S4, the note3 has a usb 3.0 port, Note 3 is on android 4.3, the better GPU is worth noting, 3 gigs of ram, battery is better and lasts longer. I mean I could say pick a plain jane fiat vs a new Jaguar, I mean all the jaguar has is better engine, features, leather, ect.. and costs way more.

True, there are some additional features, but you have to admit that the Note 3 is the "phablet" version of the S4, just like the Note 2 was the phablet S3. They come a bit after the Galaxy S release, and have some additional updates and some extra software over the Galaxy S release. But probably at least 90% is the same thing. It is somewhat like the iPhone and iPad. They are released in pairs, with one quite a bit more expensive and beefier than the other, but they are definitely siblings.

But they are not the same. My I7 extreme laptop and my i3 laptop both have windows 7 and the same basic features like CD rom, usb, ect, but the i7 is far better.