Both have AVX512 that might help, as x264 have started to use them on recent updates.
Try asm=AVX512 or asm=AVX2 or asm=AVX, but using it on unsupported CPU will instantly crash OBS when encoder starts. Most of the time, auto-detection (no "asm=") works fine.
Faster GHz speed (7940x) wins on this benchmark website.
http://cpu.userbenchmark.com/Compare/Intel-Core-i9-7940X-vs-Intel-Core-i9-7960X/m353078vsm357952
There is a limit on how well x264 can do with too many threads. Can limit x264 threads with threads=16 or less.
Edit: your gaming pc seems better then your streaming pc. The streaming PC CPU (e5 2660 v2) is kindof slow, and is without AVX2.
Edit 2:
http://cpu.userbenchmark.com/Compare/Intel-Xeon-E5-2660-v2-vs-Intel-Core-i7-5960X/m13068vs2580 This shows how terrible your old streaming PC's CPU is, compared to your gaming pc.