For the most part in current games the biggest difference will be made by the selection of the GPU. Get a great GPU + worse CPU rather than worse GPU + great CPU.
The AMD FX CPU's have many cores, which are weaker.
intel i5's have less cores, which are stronger.
The intel's consequently have better performance per core. In older games, the intels perform much better as those games are optimised for good performance with only a few cores (single-threading).
In newer games, the AMD FX's really shine due to the introduction of games using more cores (multi-threading).
The difference comes in depending on what you want to use the PC for. If you're on a tight budget, save some money and go with the AMD and spend the extra money on a better GPU that will give you better performance than any CPU could.
i5: Good for older games (single-threaded), Good for newer games (multi-threaded), Good for general work, great all-round CPU and probably the best around for current games (may change in future, see here: http://www.corsair.com/blog/ps4-xbone-pcgaming/ ).
AMD: Slightly worse for older games (single-threaded), Great for newer games (multi-threaded e.g. BF4, Crysis 3), Good for light/heavy work, extra cores are great for 3d modeling and video editing or rendering, great CPU whilst costing much less than the intel. Even though it's worse in older games it will run them perfectly well and smoothly.
Regardless, both will perform well.
AMD, Why? Great multi-threaded performance for newer games and heavy work, are just fine in older games (not overkill, can deliver smooth frame rates maxed with a good GPU), and are great for productivity with a tame pricetag.
Some non-synthetic benchmarks for AMD FX 83xx vs i5/i7:
Gaming:
http://www.youtube.com/watch?v=4et7kDGSRfc
Gaming and Streaming:
http://www.youtube.com/watch?v=eu8Sekdb-IE
Having shown that, the intel wins 9 times out of 10 in most purely gaming benchmarks as AMD cannot match its per-core performance. Also these particular benchmarks have been scrutinised many a time. If you wish to find out more you can google benchmarks for each processor.
In conclusion, budget gaming/work: AMD. Not on a budget gaming/work: i5. The i5 currently delivers better performance but don't get the impression that the AMD is lagging behind. They are great for gaming and work with a really great pricetag, just not currently up there with intel. In newer games though such as BF4 the AMD's have caught up in performance and in some cases deliver better performance than the intel's for much less money. You will get great, smooth FPS with either.
Either solution will game just fine with a nice GPU, focus mainly on that.
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