You're on the very limit, i7 is an expensive processor and usually goes in builds that are 900$ and up, you can cram it in but you'd be sacrificing the graphics card. And as I said, 7950 will bring better game performance than an i7, unless we're talking about BF4 or Crysis 3 and such. Anyway, this is what I can think of right now within the 700$ budget...
Intel build - i5 4670K, CM 212 EVO cooler, Sapphire Radeon 7950, SeaSonic 650W PSU - 699$
AMD build - FX 8350, Thermaltake NiC C4 cooler, Sapphire Radeon 7950, SeaSonic 650W PSU - 693$
I went for a SeaSonic PSU in the two above (and in my own PC) since those are quality boxes, costs the same as the one you chose if bought from Newegg. I also went with G.Skill Ripjaws X DDR3-1600 RAM modules in both builds since they are cheaper than Corsair Vengeance and not bad at all. Cooler differentiates since the AMD build allowed for 20$ more to be spent on cooling while still being under 700$. Both are formidable builds and would allow for overclocking.
OS, reinstall Windows. On Ubuntu, remove Nvidia graphics driver, comment modules you load up manually in /etc/modules (not all, just those added post-install by programs or yourself), and you won't have any issues with the hardware switch. I transferred Debian 7, Ubuntu Saucy (beta at the time) and Arch that way, no problems.
Now for the "AMD vs Intel" part...
Intel has an advantage in certain regards. Older games are a prime example, Intel makes them fly because those games use 2 cores, some 3 or 4, and Intel processors have powerful per-core performance. Skyrim uses 2 cores for heavy work and another 2 for light tasks, so an Intel CPU will run it faster due to ~40-50% more powerful core. A single Intel core will do more work per cycle than a single AMD core, that much is true. 3D modeling is also a great example, sculpting is a single-threaded process where an AMD CPU can't work as well unless it's overclocked.
On the other hand, AMD processors thrive in applications and games that are multithreaded and can take full advantage of the processor (all cores). In Photoshop, difference between an FX 8350 and i7 3770K is 1 second on a ~1m20s task, that's ~1.3% difference (in AMD's favor). Crysis is an excellent example too, where an FX 8350 @4.6GHz trails only 1-3 frames behind an i7 4770K @4.4GHz. It has problems with Crossfire/SLI in Crysis, gives only a slight improvement over a single-card setup. Skyrim, a stock FX 8350 will run it at ~75-80FPS average, while a stock i5 4670K will run it at ~110-115FPS average, so it's a bit behind Intel in that regard. Upgrade path is a good thing about AMD too, newer CPUs work on older mobos (and vice versa) since socket 939 (except APUs, those have a different socket).
Overall, both Intel and AMD will run games at a good framerate, Skyrim with VSync means that both builds will run it at 60FPS. AMD may dip below 60 in certain situations because of it's lower averages and minimums, but that's something that gets remedied when overclocking comes into play (depends on which frequency you OC it to). Some games/software that needs per-core power and/or uses a small amount of threads (1-2) will have problems with an AMD CPU, those are becoming rare but they still aren't extinct (Starcraft II for example).
So to sum up the pros and cons of both:
--Intel pros
-higher per-core performance
-lower power consumption
-mobos have good features
-newer manufacturing process
--Intel cons
-price
-bad TIM (heat)
-motherboard prices
-multi-threaded CPUs are expensive
-unlocked CPUs cost extra
-platform longevity
--AMD pros
-price
-multi-threading
-platform longevity
-motherboard prices
-no premium on unlocked CPUs
--AMD cons
-bad per-core performance
-higher power consumption (heat)
-mobos lack some newer features
-older manufacturing process
It's all up to you, weigh in the facts, what you need and what you don't, then base your selection on that. Both FX 8350 and i5 4670K are great CPUs for the price and each has it's strengths and weaknesses, the choice depends solely on what features you want/need more.
And the above two sentences are why people who buy FX 83xx yap, they expect it to perform like an i7 in every regard for a lot less money (they usually call themselves "hardcore gamers"), and when it fails to do what they want it to do, they yap like crazy. Simplified - you can't have something for nothing, some things have to be sacrificed whichever path you take. 