8700K is the more cost-effective option if you don't need extra PCIe lanes and can actually find an unit (shortage will likely persist until Q1 2018).
7820X has two more cores and twelve more PCIe 3.0 lanes at its disposal at a higher cost and lower turbo and stock clockspeeds. It can match 8700K in clockspeed if you give sufficient cooling though.
Ivy Bridge i5-3570K@4.0Ghz, Gigabyte Z77X-UD3H, 2x4GiB of PC3-12800, Sapphire RX Vega 64, Corsair CX-600 and Fractal Refined R4 (W). Kentsfield Q6600@3Ghz, HD 4850 2x2GiB PC2-6400, Gigabyte EP45-DS4P, OCZ Modstream 700W, and PC-7B.