The GS3 uses a quad-core A9 processor (specifically, the Exynos 4412) with a Mali-400 GPU while the N4 uses a quad-core S4 Pro Krait processor with an Adreno 320 GPU. In benchmarks, the S4 Pro will almost always outperform the Exynos CPU/GPU. However, benchmarks don't tell a whole lot about real-world performance.
Realistically, you won't notice much of a difference between the 2 when performing normal day-to-day tasks. In gaming, FPS may be higher with the S4 Pro but again, that really doesn't matter as long as your game works. What I'm trying to say is that the CPU shouldn't affect your decision here. The Exynos 4412 and S4 Pro are both fantastic CPUs and will handle whatever you need to get done. What should really be your deciding factor here is the design and software of both phones.
Buying a Nexus for stock Android and updates is not much different from buying a device with some sort of overlay and flashing a stock ROM or CM10. If you're worried about receiving the newest version of Android, I guarantee you that the Android developer community (specifically XDA) will get the update out faster than the manufacturer will (albeit, there will likely be a few bugs). As long as your device is well-supported by developers, that is, which is no problem with the GS3.
Many people worry about rooting and flashing ROMs for the first time when it really isn't too big of a deal. I hear people saying that flashing ROMs will "brick your device" or "your warranty will become void." At least something along those lines. Again, neither are true. Rooting or flashing a ROM rarely ever bricks a device as long as you know what you're doing and/or follow the installation guide. By flashing a ROM or rooting, your warranty is not void either. Just flash a stock ROM (and reset the flash counter if it's a Samsung device) and you're good to go.
The main reason I talked about rooting/flashing ROMs is because you seem to be deciding between stock Android and TouchWiz as well. With the Galaxy S3, if you wanted a stock ROM, there are many available over at XDA-Developers. The most popular being CyanogenMod.
Personally, I prefer the design of the Galaxy S3 because I don't want a phone that will shatter from something like a five feet drop. The S3 also feels better in my hand, just an opinion.
Your two main deciding factors should be design and software, and that is why I recommend the Galaxy S3. You have the freedom to use either TW or stock Android, and many of the other great features that come along with it. If you do decide to go with the GS3, I recommend you purchase the international version which is around $550 on Negri or $500 if you're willing to purchase an open-box version.