IndyAJD :
Here's my advice. I don't know if its a space issue, or if you guys just want it to look like a console, but the problem with HTPCs is that when things get smaller, they get pricier. I know I could lower the price of the above build by $70 just buy using a Micro ATX or Mid ATX case, and increase the overall performance. If this is an option, than I would highly recommend it as it is much more practical. I'd be willing to help you explore this alternate solution.
Pretty much what he said, though you can do an ITX machine for under $500, you might be better off with a mATX case at your budget level.
Here's my take on the $500 budget with Intel. For the money it's pretty decent and it sneaks in under $500. If you want Wifi it'll go up significantly though.
There's also builds which could feature the Pentium G3258 in an overclocking setup, but it's tricky to find the right motherboard which will support it. It's also hard to get a quality case+cooler in the budget, so I skipped over this option.
And here's my take on $500 budget with AMD. I've gone slightly over budget here but I feel like the extra cost is justified. Firstly you get Wifi due to the more expensive motherboard (there aren't really budget FM2+ boards) and the cheaper processor means you can spend more on graphics. Additionally you get an overclockable processor and motherboard - though how useful they will be in this case (and without extra funds for a suitable cooler) is debatable.
As for which is the better option, there's not a huge amount in it. The stronger graphics card in the AMD build is going to give it the legs in most games, but the Intel build isn't far behind and in general has a stronger CPU (despite only being a glorified dual core) and dramatically better power consumption due to coincidently using much more efficient components (that's just how the pricing fell).
Personally, I'd be inclined to spend slightly more and go with Intel. The upgrade path is much better, and you could feasibly build a high end system from the same parts if you chose wisely simply by upgrading the CPU/graphics.
Here's my suggestion for this. I went with the Pentium to keep the costs down but you could put anything in here as a short term solution. It has a PSU which covers you to add any graphics card you like (more or less), a motherboard which will let you add any K-series i5/i7 (as well as a load of other features), and a case which supports basically any cooler you fancy. Granted - it's a lot bigger than a console at this point but it is very flexible and could save you a lot of money in the long run.
Tyler_maguire :
Let me just start by I couldn't disagree more.... Budget = amd why on earth would you spend a 100 $ on an i3 paper weight that is absolutely crap compared to the amd 6100 which is about the same price.
Again I say that Intel will always be better than amd if you have a HUGE budget but intel will never ever touch amds value for money you cannot argue with than.
This is nonsense really, both sides have perfectly viable options at this budget level, if anything AMD have the weaker selection due to lack of FX CPU support at sub ATX levels (other than some crappy 760G boards).