I've went through a Galaxy Nexus and a Galaxy S3 within the past year...overall, I think the Droid DNA is an excellent purchase.
Pros:
1) Excellent form, feels great in the hand and I love the feel of the soft touch on the back. It's really a beautiful phone.
2) Display is gorgeous
3) Sense is stable. I haven't had one force close outside of some incompatible apps from the Play Store.
4) Signal has been great for me (although your mileage may vary based on location)
5) Battery lasts me all day but then again I'm typically always near a power outlet. However, I never felt anxiety over having a depleted battery when I'm out and about. Again, your mileage may vary.
Cons:
1) Camera has been disappointing but average on a smartphone camera scale. Pictures turn out noisy for me on auto but even when I play around with the camera settings, I never really get great results like I did on the GS3. On that point, the GS3 wins. I'm hoping a software update can improve image processing a bit because it could definitely use some tweaking.
2) The display might be a bit too tall. I find myself having to really stretch to pull down the notification tray or reach action bar items that are positioned at the top of apps. However, I knew the display would be big going in and it doesn't really bother me too much. However, I think 4.7 inches is my sweet spot for one handed usability (I never had issues with my GNex in this regard).
3) Sense is rather ugly to me but that's because I prefer the look of Android's stock UI. If they could incorporate the functionality of Sense with the look of Google's own Holo UI, then I would be a very happy man. I also think they could have done away from the dedicated capactive buttons for navigation and went with on-screen buttons like on the GNex. The screen is certainly tall enough to incorporate it without using too much screen real estate.
4) Battery door ripped off in just a week
But as been said before, you really need to use the phone in person to get a feel for it. Also, I don't feel it's fair to compare this phone to the Note 2 as they are in slightly different categories. I'd say the main head to head for the DNA is the GS3.