According to this site, Sony NEX F3's sensor is 3x as large as Nikons, and it's image quality is 73 vs Nikon's 56 (how on earth do you come up with a number for this?). I have been experimenting with both cameras by taking pictures under low light settings and landscape settings, yet I don't see my Sony performing better than my Nikon. What's a straight forward way to show the power of NEX F3's large sensor and its superior image quality?
In theory, there should be two areas where the larger sensor provides a noticeable benefit:
In fact, it is the case that the Sony NEX F3 goes up to ISO 16000 while the Nikon 1 J1 tops out at 3200 (with a 6400 "boost" mode). You can look at pixel-level crops from test images from reviews like this one at dpreview where it's visibly clear that the ISO 6400 images from the Sony are more clear, particularly showing less chroma noise). But, to my eye, the Sony images just one step up at ISO 12800 are worse than the Nikon ones at 6400, so it's not like they're on the other side of the world from each other.
You can also see more in the DPreview test images at lower ISO, and the resolution numbers in the two reviews (1 J1, NEX F3) back that up. The 1 J1 review specifically notes the Sony NEX cameras as having "significantly higher" resolution. If you're making photographs where detail is important and you are using a tripod and high quality lenses and you are making large prints, this is an important consideration.
Additionally, there are some other intrinsic effects. Assuming you print at the same size, images from the smaller sensor will have more depth of field. (More will be in focus.) This isn't about image quality, but it may be something to think about when looking at the different systems (and especially when comparing lenses).
Since the sensors used in these cameras are both very good, in many conditions, you won't really be able to tell which is better. If no one can reasonably tell the difference in image quality, for all practical purposes it doesn't exist. Therefore, you should make the decision between these cameras based on something else. Ergonomics is a big one, of course, but the most important of all is the lens lineup for the system. Currently there are more NEX lens options than NIKKOR 1, but in both cases the selection is relatively small, and it may be that the smaller system happens to have just what you want.
This is a huge flaw in the Snapsort comparison you link to. "2x more lenses" doesn't mean a thing if the lens you prefer is in the smaller set but not in the larger one.
The "image quality" comes from DxOmark's technical measurements. The lens number is just kind of silly, but it is horribly irresponsible and misleading of Snapsort to represent the DxOmark number in the way they do. You'll gain a more useful understanding by actually looking at the charts on DxOmark, but still should take it with a grain of salt. Particularly, the image quality of all cameras in this range is on the order of amazing to excellent. The scale inflates the difference, making it look like the issue is C- vs F, which is completely ridiculous. It's true that one can (even without pixel-peeping) see some quality diffences between various cameras these days, but taking the score (or even the technical tests in all of their detail) in isolation is not helpful. Taking the DxOmark number and coming up with the claim of "30% better image quality" can really only be described as bull excrement.
(I should add that it looks like they've changed the wording of another technical measurement, color depth. Here, it says "distinguishes 1.2 more bits of color", where previously I believe they auto-translated that into a dramatic but practically meaningless claim like "distinguishes 3.8 million more colors". I once was in a cave in Germany which claimed to have "the most colors of any cave in the world". Turns out they were all boring browns and grays.)
Anyway, complaints about Snapsort aside: you can clearly see the difference in test samples in any review. However, both cameras are excellent and both are lauded for their image quality. Unless you have specific, special needs, this should not be your deciding factor.