I had AT&T Note 3 for couple months now and tried unlocked developer edition Moto X for about two weeks.
I prefer the Moto X, but had to return it because NFC was not working, and they would only replace it with a like-new-used device. I'm actually waiting for the Moto X2 now.
Now here's the long version of my thought on both devices.
Pros of Moto X:
Active Display is pure awesome. Every time I reach to my phone, I can see the time, and that's what I need it mostly for. To unlock, I don't even have to press the power button.
Moto X's smaller size makes it easier to carry in my pocket. Every time I get on the drivers seat, I have to take out Note3 from my pocket because it's really hard to pull it out while sitting.
Everything feels faster than Note 3. I've never used task switcher this much, and it was very smooth too. Note 3's dialler sometimes takes 2-3 seconds. Sometimes when I try to open Contacts, it tries to connect to AT&T Contacts and takes another 5 seconds. Wow, is Moto X really just a dual core Snapdragon S4 Pro? And Note 3 has Snapdragon 800 quadcore but it's slow [inappropriate language removed by moderators]
Shake to launch camera is also pretty quick and useful.
Cons:
Battery life is not that good. I use up 40% in just 2 hours moderate use (no games).
Camera quality below average. RGBC pixel fail?
There's a hairline crack problem on the back piece of the case especially on the developer edition.
Pros of Note3:
Digitizer is really useful. The large screen is a must for using digitizer pen, and does give better experience for web browsing.
Great battery life, removable battery and microSD card. It feels like with Moto X I get about half of what Note 3 gets.
Much better camera, but slow to launch.
Build quality is pretty good, and don't need a body protector on the back because it's scratch resistant.
Cons:
AT&T version has too much bloat. Seriously too much from both AT&T and Samsung. I had to disable 50 apps, and then there's still more that can't even be disabled.
Locked bootloader makes it unrootable, you are stuck with AT&T crap.
Every time you try to tether, it has to connect to AT&T and verify that you have a tether plan. This adds additional couple seconds of time just to launch the tether app.
Poor overall experience. Sometimes it takes 2 seconds just to see the time after pressing the power button. As mentioned earlier, dialer app is slow, contacts app is also slow and half baked. S-Voice sucks.
Too much gimmicks and unpolished software.
Too much crap bloat. I went to play store and rated them all 1 star.