You can buy hardware with lower-power Intel chips from NewEgg, you just can't buy the CPU by itself. It's not in your price range, but here's a Zotac mini computer for $380 (which includes the RAM and SSD): http://www.newegg.com/Product/...
They also sell numerous tablets/laptops/etc with that processor.
For an Intel motherbaord/CPU/RAM combo, you're looking at around $40 for the motherboard (all these prices from NewEgg US), $30 for 4GB of RAM, leaving you up to $130 for the processor. That puts it at an Intel i3-4360, a high-clocked dual-core Haswell part.
For AMD's part, assuming the same CPU budget, you're looking at an A10-6800K.
Right off the bat we can notice a disparity in TDP, being 54W on the Intel chip and 100W on the AMD chip.
In terms of performance, AnandTech Bench conveniently has both of those chips in their system. The benchmarks show that the Intel chip is faster (sometimes substantially) in almost every benchmark, be it single or multithreaded... And all that while using much less power.
After looking up those results, though, I realized that that AMD chip was a Richland, while there is the newer Kaveri cores available. There is also a Kaveri CPU at the same $130 pricepoint (but with a lower clockspeed/model number), the A10-7700K, with a TDP of 95W. That one is unfortunately not in Bench, but looking at other sides indicates that it manages to narrow the gap substantially, while still generally being slower than the i3 chip. But in order to do this, it uses 10W more power at idle and 26W more power at load.
Intel's got Haswell chips at decent clockspeeds down to roughly the $40 pricepoint, where I expect they'd still compete favourably with AMD. Below that is nothing, and I suspect there might be a point somewhere between $40 and $130 where AMD makes more sense.
All this said, I'm disappointed in AMD. I don't want Intel to beat them, I want AMD to put out parts that are competitive in price, performance, AND power consumption. Most of my early CPUs were AMD chips. I've owned a K6-2, a Duron, and an Athlon XP, and all were fantastic. The Athlon 64 was also great. But ever since then, AMD has been behind, and the lack of competition has certainly not helped the market. I keep hoping that AMD will put out something new that wows me, a completely new architecture that shakes things up like Conroe did for Intel or ClawHammer did for AMD. But thus far they seem to keep iterating on the same non-competitive designs.
Unfortunately, it seems like ARM is more likely than AMD to bring competition to the market as they keep slowly creeping up the TDP ladder, and we're just now starting to see them going head-to-head with Intel in the PC space with lower-end Chromebooks. nVidia's Denver core is supposed to be competitive with Haswell in terms of performance/power, but I can't find any benchmarks that directly compare them.