I had the same 3 phones on my shopping list, and got the Note 3 a couple weeks ago. I'm still figuring it out but so far it has exceeded my expectations.
Really, all 3 seem like decent phones, none is a bad choice.
I ended up selecting the note 3 because I never talk on it; I pretty much use it just for internet. For that purpose, the larger screen size and pen make a huge difference in how usable it is. The split screen mode where you can use 2 apps at once is really helpful too.
The pen gives you the precision you need for stuff like remote desktop or accessing non-mobile versions of web pages. This can be super-frustrating to do using your fingers on a regular phone or even a tablet, but it actually works OK with the pen. The hover function of the pen lets you see where you're going to click before you do it, rather than just poking the screen with your finger and hoping for the best. If you use remote desktop, get the Microsoft app for it since it supports the pen properly.
I was hesitant about the size of the note 3 at first, but it hasn't been an issue so far (however it's winter and I'm always wearing a coat with giant pockets).
I haven't messed around with ROMs yet but in my research it seemed like the T-mobile variant of the phone was the most flexible. There are 5 or 6 different note 3 variants, some of which support 4G/LTE and some that don't.
Also, there's a Chinese knockoff note 3 (labelled with Samsung branding and packaging and everything, looks almost identical, but with substandard hardware inside) so watch that you don't end up with one of those through a shady craigslist deal.
