I own both, and if I was only able to have one...
....I would have to pick the Note 10.1
It is hard to explain why, but if I had to generalize - it is because it is simply more interesting and, for me, is much more usable as a daily driver than any other tablet. I am lucky to own several tablets: Xoom Wifi, Nexus 7, Nexus 10, Galaxy Tab 7 7.0, Galaxy Note 10.1, and iPad 3 32GB/3g. I have also owned the Lenovo K1 10.1" and the Transformer Prime along with a rooted Kindle Fire with JB on it.
I have owned them all within weeks of their launch. There is no question whatsoever that the Nexus 10 is faster than the Note. I base my performance measures using Octane Benchmark on Chrome Beta which benchmarks on Octane than any other browser that I have found. (The Note averages a little over 2,000, whereas the Nexus 10 averages a blazing 3,800!)
The JB 4.1.2 update for the Note has made it even more useful - a smoother experience for certain. I actually like Touchwiz after the JB update. To me, the Nexus 10 seems just plain now. I never thought that I would say that as I was always striving to get the pure Android experience.
I like that Samsung still supports Flash in their 4.1.2 build (in the default browser). Flash is still a must for a 100% web experience, though I am not a huge fan.
I do like the Nexus 10, and was very happy that Samsung, not Asus (!), was building it for Google. My past Asus experiences were terrible (I try and forget the Transformer Prime) and though I had high hopes for the Nexus 7, it has proven to be solidly an Asus product - laggy - feeling little, if any faster than the Galaxy Tab 2 7.0 with the recently released JB 4.1.1 update on it.
I think the ideal tablet would be a cross between the two devices - perhaps that would be the perfect Android device I have been waiting for (no, unfortunately, none of the ones I have now can be considered that perfect one). If the Nexus 10 becomes the basis of the next gen Note, then we may just may have the perfect device.