That's a beast of a computer, very nice. Personally I'd go for the 1st over the 2nd but that's just because I prefer Intel, although I don't know how they compare nowadays.
Also DDR4 is still pretty, and I don't know if it's worth getting it just for gaming. Quantity of RAM is still more important than clock speeds, so 16GB of RAM will be plenty, even in DDR3.
You could spend the savings on a HDD to store games on if you want, but I don't know how much space you'll be needing. Games are getting very large (50-60GB for some new releases), and I don't know if 500GB will last if you want to only use an SSD.
You'll need Windows as I don't see it mentioned being included anywhere, but basic wireless capability is inbuilt in motherboards. If you have particularly rubbish WiFi (in apartment complexes or something) you could get an extra wireless card or a USB dongle but it's usually not needed.
I assume you'll be reusing your monitor, mouse and keyboard?
You'll need to get Office too, or use a free programme like OpenOffice, but it's not as polished (it is completely compatible though, so it does work as a temporary solution).