EDIT: It's probably a good idea to lay out some ground rules at the top so people know how I'm approaching this:

1. Everything must be new. The reason being the used market is not consistent or reliable.
2. Everything must be gotten from official channels.
3. Hardware must be matched as closely as possible. If the exact part or equivalent cannot be found, a justification must be needed.
4. If a part on the PS4 is available for the PC, it must be included.
   1. If I'm allowed to leave out some parts though, its price must be reflected against the console. So if I omit a $40 part, I must subtract $40 from the PS4's price.
   2. Certain things that are not known like the secondary ARM chip and its DRAM, can be left out for the sake of argument.

Every once in a while I like to see if I can build a machine that closely matches the PS4's specs as much as possible and see how much it would cost. I've done the PS4 a few times, so let's do the PS4 Pro:

PCPartPicker part list: https://pcpartpicker.com/list/g9L6pb
Price breakdown by merchant: https://pcpartpicker.com/list/g9L6pb/by_merchant/

CPU: AMD Athlon X4 880K 4.0GHz Quad-Core Processor  ($92.34 @ Newegg Marketplace)
Motherboard: ASRock FM2A68M-DG3+ Micro ATX FM2+ Motherboard  ($44.99 @ SuperBiiz)
Memory: Crucial 4GB (1 x 4GB) DDR3-1600 Memory  ($24.88 @ OutletPC)
Storage: Toshiba 1TB 2.5" 5400RPM Internal Hard Drive  ($54.89 @ SuperBiiz)
Video Card: Gigabyte Radeon RX 480 4GB G1 Gaming Video Card  ($189.99 @ Newegg)
Case: Rosewill RANGER-M MicroATX Mini Tower Case  ($24.99 @ Newegg)
Power Supply: SeaSonic S12II 430W 80+ Bronze Certified ATX Power Supply  ($49.99 @ Newegg)
Optical Drive: LG WH14NS40 Blu-Ray/DVD/CD Writer  ($46.88 @ OutletPC)
Total: $528.95
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-12-09 13:02 EST-0500

Most of the parts were the cheapest parts I could source that at least wouldn't blow up in my face (hence the SeaSonic PSU). But let's go over the key points here:

• Processor: The PS4 Pro has a 2.1GHz 8-core Jaguar. In this I included a quad-core Athlon X4 running at 4.0GHz. The assumption is that doubling the speed but halving the cores will result in similar performance. Also the Athlon X4 uses the Steamroller architecture, which should be a bit better than Jaguar's
• GPU: The PS4 Pro has something that resembles a downclocked RX 480 (911MHz vs 1120 MHz). So I went with the 4GB version of the RX 480 mostly because from the only resource about GPU memory consumption I could find on the PS4, graphics consumes about 3.5GB.
• Memory: 4GB for the GPU leaves 4GB for the system to keep it "fair". That same resource also suggested that 3 GB or so was used for the system outside of the game.
• Storage: The PS4 Pro comes with a 1TB 5400RPM 2.5" drive, and probably a cheap one too.
• Optical drive: Because the PS4 Pro has one and I need to keep this fair. But it is the cheapest one PCPP had listed.

Hm... Even without an OS or peripherals I'm still about $130 over and there's not a whole lot I could cut off.

EDIT: As of 01/04/2017, the price jumped up another $36 for the parts, to a total of about $566

PCPartPicker part list: https://pcpartpicker.com/list/g9L6pb
Price breakdown by merchant: https://pcpartpicker.com/list/g9L6pb/by_merchant/

CPU: AMD Athlon X4 880K 4.0GHz Quad-Core Processor  ($92.29 @ SuperBiiz)
Motherboard: ASRock FM2A68M-DG3+ Micro ATX FM2+ Motherboard  ($44.99 @ SuperBiiz)
Memory: Crucial 4GB (1 x 4GB) DDR3-1600 Memory  ($23.48 @ Amazon)
Storage: Toshiba 1TB 2.5" 5400RPM Internal Hard Drive  ($54.89 @ SuperBiiz)
Video Card: Gigabyte Radeon RX 480 4GB G1 Gaming Video Card  ($224.98 @ Newegg)
Case: Rosewill RANGER-M MicroATX Mini Tower Case  ($33.30 @ Jet)
Power Supply: SeaSonic S12II 430W 80+ Bronze Certified ATX Power Supply  ($44.98 @ Newegg)
Optical Drive: LG WH14NS40 Blu-Ray/DVD/CD Writer  ($46.88 @ OutletPC)
Total: $565.79
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-01-04 19:11 EST-0500