I had a hard time choosing between these two as well as my last phone was the S3 but I wanted something a bit larger. The Note 2 is a bit too big to be a phone for my liking. I finally decided on the DNA and I will list my reasons with backing rather than just a quick answer on what I THINK is the best phone. I went to the Verizon store this morning and grabbed a brand new Note 2 for comparison. First would be the display. The DNA's Super LCD 3 at 440ppi is so much nicer than the Note's SAMOLED at only 267ppi. The difference in the sharpness and clarity of photos, videos and even text on the DNA is huge. Then there is the processor. The S4 pro easily beats out the Exynos in the Quadrant and benchmarks as seen in my pics below. My DNA scored an 8127 in Quadrant and a 21223 in Antutu compared to the Note's 6184 and 17580.
My next reason for choosing the DNA over the Note 2 is build quality. Although being better build quality, the DNA is still a good bit lighter than the Note and its size is more like a big phone rather than a small tablet. I am not a fan of the Note's (or my S3s) physical home button. There is no reason for it, make it capacitive. Another pro for me is the great camera and features of the DNA. Not only can you edit the pic with about 20 built in effects that enhance the photo, you can also then change the white balance, levels, exposure, contrast etc...these are features most cameras dont even have and you need to drag the pic into an editor to get. HTC did a wonderful job here and although the Samsung cameras are good, the quality plus the features on the DNA are just above and beyond anything on the Note.
Another big plus for me was the way Sense integrates social media and your contacts. Go into a contact on the Note and you have the standard things. You can call, text, email, add to a group and such. However, go into a contact on the DNA and aside from all of the usual interactions, you can also start a chat with Google Talk and view their Facebook and Twitter profile that shows their latest post. Then you can click "thread" and it will bring up all of your messages, emails and call history with them. Slide over to the "updates" tab and you can see all of their recent Facebook and Twitter updates including photos and then finally you can click the "gallery" tab and see all of their Facebook photo albums. Again, this is just from their contact page! Outstanding social media integration unmatched by Samsung or TouchWiz. This is far more useful to me than a pen to doodle around with.
In the end the fantastic display, superior processor, build quality and Sense's ability to integrate amazingly well with social media and be highly customizable with different scenes and skins led me to choose the DNA over the Note. There are things I like about the Note 2 as well. Excellent battery life, removable battery and storage, multi window and I like the phone part of it and how it manages your recent calls and dialer better. Much less clutter going on. I dont really need all the storage as I have an iPod Touch with all my music and I am not one to watch HD movies on anything less than my huge TV in my room
But its something to consider if your life revolves around your phone and you have to do everything on it. Hope this helps with others you are on the fence, both are great devices and you really cant go wrong with either.