In my personal opinion, there's not nearly as much money to be saved now-a-days by building your own computer over buying one pre-built. At most, I would say you would save $200 if you spent a lot of time shopping around (different places) for equivalent/cheaper parts. Another difference to keep in mind is that if you build your own, most likely your parts will not be under any type of warranty (I'm not even sure if the build you linked to has a warranty on it, I only saw the specs). One last thing I would like to mention, in regards to gaming is the HD. While 1 TB is a lot of space, having a SSD would create a noticeable difference in load times (especially when zoning, etc). The only down-side is that you're trading some real-estate for the performance boost if you switch it out.
I'm sure your friend won't care much for any difference, as she'll be happy being able to play on whichever build you choose. I just wanted to point out that the SSD is often an overlooked feature for speeding up online gaming.
I'm sure someone with a little more experience on specific parts may be along soon to clarify or add additional information.