If you read my Late 2016 PC builder's guide, I heavily discouraged going with AMD. AMD's FX series is the latest full CPU launched from the company (Kaveri APU's came in 2014, but an APU usually doesn't work as well as a full CPU) AMD's latest FX CPU is four, nearly five years old. But as much as I discouraged them for building, they might not seem as terrible as you might have thought. I own an FX 8350 at this point in time and it does everything it needs to without capping out or causing any inconvenience to me. That said, a lot of stuff I run was programmed before or not terribly long after the FX 8350, save for games like Doom, Fallout 4, and Overwatch.
Our comparison CPU will be the Intel CORE i3 6100, since it is the most relevant CPU at it's price point.
1. Benchmarks ; An elitist's measurement tool
You might look at Benchmarks online to find how well the CPU you might be interested in performs versus others. On userbenchmark, my FX 8350 checks in at 145 out of 913 CPUs tested on that website. I'm not surprised, or feeling like I need a new CPU. Why? Because LOADS of the CPUs at the front of the line are Intel extreme series CPUs, CPUs which cost hundreds of dollars and are enthusiast, workstation, or server level.
Am I saying my 8350 is lightning fast? No. That doesn't change the fact that it's platform is outdated. But I am saying that you aren't gonna be looking at a benchmark all day. If you need the extra cores and threads of an 8350 over the i3 6100, and you fix and forget your PC, then it's still a fine choice. The benchmark doesn't make a CPU bad. What you're looking to do with it can, it depends on YOU.
2. Performance vs Price
Intel CORE i3 6100 is available from a number of PC stores around $120 USD, from where I live. FX 8300 checks in at that price, 8320 for 130, 8350 for 145, 8370 at 180, and 9590 at 200. I'll discard the 8370 because it's basically a slightly more streamlined FX 8350 with more ability for overclocking, and the 9590 because the requirements for a motherboard are very high (220W TDP at stock speeds, mainly)
At stock speeds, the i3 6100 beats out all of these up to the FX 8370 on the benchmarks. But there is something that simply the i3 cannot do ; gaming with more than two cores. The FX 8350 has plenty of cores for gaming, which is why it makes the cut for doom but the i3 doesn't. The 8350 also has many more threads to overall be able to crunch down more tasks at a time. Furthermore, nearly every AMD CPU has something under the hood that most forget about, but it can be easily put to use.
3. Overclocking
The Intel CORE i3 6100 has a locked multiplier ; any overclocking needs to be done with the bus. Furthermore, I believe Intel's chipsets disallow overclocking until the Z170 chipset (I could be wrong, it might be multiplier only overclocks)
AMD FX CPU's are all black edition; that signals that the CPU has an unlocked multiplier, meaning it could be overclocked as far as the multiplier can go, or really, as far as you feel like taking it. As long as you have a decent aftermarket fan, or better yet, are using AIO or custom water cooling, any 8 core AMD FX CPU should be able to overclock to beat out the i3 6100 (short enough that a nice air cooler like Cryorig H7 or Cooler Master Hyper 212 EVO can get it) And, if you overclock right, the noise level or operating temperature should not exceed what is normal either way. The result is a much more powerful CPU for gaming and multitasking at a price that Intel simply cannot touch, which is very impressive given it's age.
And now ; Where the CORE i3 6100 beats the FX-8XXX series.
Operating Temperature
This point is totally invalid if you have a decent aftermarket CPU cooler. The only cooler that I have tested my 8350 on where it pushed beyond it's safe temps was the stock cooler, AMD Wraith. You shouldn't expect too much from a stock cooler anyways.
Either way, the safe limit for the FX 8350 is 62 degrees C, whereas the 6100 can go up to 65 degrees. However, the i3 also has a newer manufacturing process which makes it less likely to get there. My point still stands ;
this whole point can be discarded if you take some time to get quality cooling. A good cpu can can be had for under $40 easily (Cryorig H7 takes the gold, Cooler Master Hyper 212 EVO is also quite nice, Cryorig M9a is great under $20, loads of low profile options too)
Power Consumption
This is actually an arguable point. Base power consumption of the i3 is about 45 watts, whereas the FX 8350 needs 125 watts. This forces a bigger power supply, more if you wish to overclock, but a nice quality 650 watt power supply can be had for $60 (Corsair CX650M) With more power being drawn, you will have to pay more on your electrical bill truth be told. Still though, I don't think a 20 dollar difference would mean too much to me.
Upgrade Pathing
Here is the best reason to choose an i3 over vishera at this price ; Upgrades. The FX 8 core CPUs are top of their product line. There's nowhere else to go but overclocking. The i3 utilizes the LGA1151 CPU socket and can be swapped to any CPU with a TDP that the board can handle, including any other Skylake CORE i5 or i7 CPU. Intel's Kaby Lake upcoming was also rumored to be able to run on the LGA1151 socket.
Counterargument ; If you're a budget person, you might not always be looking for upgrades. Otherwise this is a point that should be taken into consideration ; the only upgrade as far as I know is the FX 9590 which is basically comparable to an overclocked cpu of the 8 series pretty much. (But since it's stock clocks are faster, it can go further, you just need to have really good cooling)
Built in GPU
Generally the Intel Graphics HD 530 probably won't make much of an impact on people looking for game performance as even a budget gamer should own a 2GB video card that works in a PCIE x16 slot I hope. The only thing said GPU would be good for is a low price office desktop which won't need a beefy GPU to run things. Furthermore most motherboards will have an ATI Radeon integrated chip likely installed for onboard video support anyways, if you demand that hardware (can be handy for troubleshooting)
Motherboard cost isn't worth it
If you want to actually overclock your FX processor, and it's an 8 core model, the issue comes when you need a decent motherboard that supports it. 8xxx and 9590 are the high end CPU models, and usually force you into the 990FX chipset models. Personally I only trust Asus and Gigabyte on the AM3+ socket, and that traps us into 100 dollar motherboards.
See? The UD3-R5 was also the least expensive of all 990FX chipset boards... basically my point here is that it costs a lot just to get a good board to overclock on, which kind of takes away of AMD's low price appeal. An Intel Z170 motherboard would cost somewhere of the same and you could suck it up and pay $100 more for an i5 6600k CPU.
Now, that said, if you're fine with stock speeds, some of these 8 core beasts can run on one of my favorite budget boards ; Gigabyte GA-78-LMT-USB3. It's really freaking old technology honestly (Sata II, DDR3-1333, 760G chipset) however it's $50 and supports AMD CPUs that have 125W or lower TDP. Of the 8 core Vishera models ; FX 8300, 8320(E or regular), 8350, 8370(E or regular)
Of those, I'd suggest 8300 ($120, 3.2 GHz) or ideally spend the extra 30 bucks and pick up an FX-8350 (4.0 GHz) and it pairs well enough with the board. I do, however, suggest having good case cooling as the VRM temperatures could potentially be a problem with that board if there is a lack of good cooling.
Final thoughts
If you want the best performance possible under $150, then an 8 core FX CPU (ideally, the 8350) will perform much better at multitasking and better overall when overclocked thanks to the sheer amount of cores and threads, as long as your cooling is nice. It only gets better with age.
For the patient / richer
AM4 CPUs are expected to release before Spring 2017, so if you can get by until then and have the extra cash for an AM4 processor, Summit Ridge is expected to blow the FX 8xxx series away but will likely come with a hefty price tag as well as the appropriated motherboards and DDR4 RAM that will come with it. Again, if you have the "munz" then it would be worth the wait.
SHOW ME THE BUILD!
I gotcha pal.
At 550 USD we get the 8300 on a pretty nice budget setup, including a video card. Not good for overclocking and you'll have the AMD Wraith but if you just wanted to beat an i3 gaming type setup then this will suffice.
At 800 USD we can get an easily overclockable AMD rig. The issue really comes down to the price you pay ; you'll beat an i3 but maybe not a midrange i5 setup. You could lower the PSU a bit maybe, swap out the H60 for a cryorig H7, down the motherboard to 990FXA ud3 r5 as they don't do too great past 4.5 GHz on the 8 core setups, which could bring you closer to 700 USD.
For this price I'd just get an i7 6700k and stuff. Not worth it for the 9590.
But as you can see, my point still stands ; if you are a lucky owner of a good 990FX motherboard, the simple upgrade that can do a number on performance is an 8 core AMD FX overclocked a fair bit. If you need the best rig possible under 550, chances are it's gonna be the i3... but an 8300 can crush your multitasking needs easily.