At stock clocks the 8400 will be quite a lot faster at ST and games due to a combination of higher IPC and clockspeed. MT performance would probably be slightly behind the R5 1600 due to the lack of HT.
At default the R5 1600 has a base clock of 3.2GHz for 12 Threads and 3.6GHz ST turbo. The Gaming performance difference should be very close (5-10%).
Of course you can overclock an R5 1600 to approx 4GHz to close the ST gap and extend the MT lead, but at a significant cost to power consumption, like any other CPU that is pushing its clock speed ceiling with raised voltages. I wouldn't be surprised to see a 1600 @ 4GHz exceed 150W power draw (depending on voltages). Maybe power efficiency suddenly doesn't matter to AMD fans though
Overclocking the R5 1600 to 4GHz all 12 threads will make it 50%+ faster in MT than the 8400. Yes it will consume more but it will make it a lot faster too.
Also if you really like lower power consumption then you just OC to 3.8GHz and you still have way higher MT performance than 8400 but very close ST performance as well at reasobly power consumption.
8400 will be very popular for OEM SFF desktop PCs with iGPU only, DIY builders and gamers will chose the R5 1600 and 8600K.