Just speaking between the FX or A10 there is simply no comparison because the FX is really the only choice if you have to choose. As for the Intel choice I'll let someone else comment on that comparison.
For Amd get an 8320 it will beat the a10 any day, for intel a 3570k, or 4670k. PCIE 3.0 does not matter, especially for a gpu under 300 as 2.0 x 16 should be plenty fine. As for gpus at 160 usd gtx 660, 180-200 usd r9 270 if possible, and at 240 usd get a gtx 760.
Again on the cpus get intel if you can afford it in my opinion it is the better choice at the moment. 8320 is around 150 usd and 3570k and 4670k go for around 200 to 220.
APUs (A4, A6, A8, A10) are built for low power, low cost, low 'demand' uses (i.e. Grandma getting to Facebook or playing Flapping bird) by just opening the box, turn and use the PC. GAMING on AMD is only in the FX class of cards, and as the 2013 games showed only the top of the line FX-8xxx CPUs can be the only competition to Intel, until Haswell was released. Buying a APU is (as you noted) far less performance and when investing in higher end video card (7xxx, Rx, Titan, etc.) becomes the bottleneck for the GPU which is sitting around twiddling thumbs.
Now to caveat here, yes we are talking scoring of say a i7 does 98fps on Ultra/1080p/64Map with a Titan as compared to 93fps of a i5 as compared to a FX-8xxx of a 90FPS as compared to a i3 getting 86fps. But the point is not just BF4 today, but BF5/6 tommorrow and going forward that those 'close numbers' really changes dramatically. As in the case of those whom invested in FX-4xxx or FX-6xxx on BF3 when compared to i5 or i3s were similiar 'close' numbers, but now on BF4, FX-6xxx are 'maxing' out the CPUs and FX-4xxx can't seriously play the game. The better 'extra cost' investment back then now allows the i5 and i7-2xxx or 3xxx able to play BF4 properly, though I doubt when BF5 comes out they will be AS capable (dropping graphics down to low-med).
The main difference as you pointed out is, at this time, and per all accounts, AMD is NOT investing further in CPUs, but instead shifting the entire business to the original goal (when they bought out ATI) of a All Processing in one Unit (APU) combined GPU and CPU for the MOBILITY marketspace (read that as your smartphones, tablets, phablets (fat tablets), and low end laptops). Nothing else was introduced by AMD at E3 nor CES (the BIG event shows to show off new techs).
Considering the huge performance increase (should look up i3 Haswell BF4 on youtube, OMG!) Haswell has brought to the table, I would recommend Intel over AMD. While previously the costs comparisons were enormous, Intel has further forced a immediate price drop on the Haswell line on 'replacement' systems (individual parting is more expensive normally). For example both Acer and Lenovo have had a i7 Haswell, 8-12GB, 1TB, DVD, W8 systems for only $599. Add to either a good PSU plus a decent GPU (Nvidia 760 is only $249 for example) for additional $200-300 one time cost, you get a serious rig doing the Ultra/1080p/64Map above 50FPS to 80FPS. REALLY not that costly considering how new the Haswell is much less how historically a i7 was quite expensive.
Now I will also say, MANY people are balking at $700 gaming PCs, or GPUs (nothing else) costing $150, 200, 300 or more, and is a significant reason people instead spend less then $200 for a PS3/360 which has played all these games and more quite nicely for over 10 years, and even new games (even though Xbox One/PS4 is out) are also for these system still with no worry about extra 'costs' for GPUs, which CPUs to buy or anything else. Most are looking at it, I only got to spend one time $500 for a NexGen and 10 years from now still have a kick ass system to play on. Just food for thought.