Obviously the following is all opinion, but take from it what you want... I can only speak to video, text, brightness and color. I can't elaborate on outdoor visiblity or pictures but online reviews say that the G Pro is better in outdoor vis, and depending on who you read, it's a toss up on the camera, with some proclaiming one is better than the other, while others totally disagree (I wonder if different builds might use somewhat different sourced camera components).
I Was set to purchase the Note 2 due to all the positive press it received. I so wanted a phone with a stylus (I miss my Dell Axim x51v). I even picked up a Note 2 demo phone to carry around for a week or two to see if it was too big or heavy. I bought a case and screen protectors for it in anticipation of actually getting the phone.
Then, I went down to AT&T store to play...I was not happy! Even after brightening the display to max, setting the color temp of the screen (forgot the proper terminology), and setting the brighness for the browser, the screen to manual (auto was too dull), I found the screen to have little to no white coloring. At best, it was a pale blue color. When comparing it to my 4 year old iPhone 3GS, the Note 2 was about half as bright as my iPhone. I realize that my phone has a LCD while the Note 2 has an AMOLED, with it's boosted blues, reds, and greens. I can live with the super saturated AMOLED colors, but i cannot live with the absence of whites. The G Pro has a screen that is similar in its characteristics to the iPhone screen, only it's brighter, and had much higher resolution.
When trying to read a webpage for example, the difference is clear...the Note 2 with its relatively low pixels per inch count gives you jaggy lettering in comparison to the IPS LCD of the G Pro (and my old iPhone). In regards to the contrast between the black lettering and the white screen, you might figure that the Note 2 would win due to the fact that AMOLED has better blacks than IPS LCD (AMOLED uses no power to make black, it is black due to no pixels being lit. LCDs always have a white light on behind the panel, so even though you might want black, you still have some minimal amount of white bleeding through, kind of making your blacks somewhat more of a gray), that is not the case. The grayer text of an LCD looks better, at least to my eyes, due to the much, much brighter white screen that the G Pro has. The Note 2 has jet black text, but its set off above a not all that white screen, which makes the G pro better for reading, for me.
In comparison when viewing YouTube video...When viewing on a Note 2 alone, all by itself, it looks fine and most might be happy with it. That is until you simultaneously view video on the two devices side by side. Then you will really see what you would be missing with the Note 2! Although not as contrasty as the Note 2, the G Pro gives you more realistic color, a much brighter screen, and a huge amount more of definition due to more PPI. I was watching a World Rally video taking place on dirt roads, on both devices, with the cars spitting up clods of earth, while cornering past billboards and impromptu road signs. On the Note 2 the kicked-up dirt appeared as a cloud of smoky dust. On the G Pro you could make out individual clods of dirt and pebbles and such. On the Note, as the cars passed the signs, the lettering was blocky and jagged, but on the G Pro, the lettering was smooth with no case of jaggies. To me, the G Pro gives a much better YouTube experience. The Note 2 has a PPI of 267 or so, while the G Pro has a bit over 400 PPI. It does make a difference.
I couldn't gauge the difference when taking pics in the store. Both seemed to do an admirable job in that store, on that day, with that particular lighting, and viewed on thier own screen. But, as the phones were chained to the wall, i couldn't take them outside to see the differences in bright daytime lighting. I also could not compare the photos blown up on a common PC screen.