The issue with the 8300 is that at stock its a weak performer when it comes to multi threading.
You can look at the 4.2ghz turbo speed & comparisons on CPU boss etc & assume its significantly stronger than a 6300.
Look into it further & you'll find at stock speeds the following.
The 8300 can hit 4 GHz on 4 cores while lowering 4 remaining cores to 3000mhz OR 4.2GHZ on 2 cores while lowering 6 remaining cores to 3000mhz.
3000mhz is a big big drop if a title is multithreaded, if all cores are symmetrically loaded the whole chip will just sit at the base 3.3ghz.
Fx 6300 - 4.1 GHz on 2 cores , 4 cores drop to 3300mhz OR 3.8 GHZ on 4 cores with 2 cores dropped to 3300mhz.
Bear in mind the above is at stock settings but generally in gaming (especially with amd chips) core speed is key performance wise
You can just disable turbo & set a 6300 to 3.8ghz on a stock cooler with absolutely no voltage or temp increase over stock settings.
That is something you just cannot do with an 8300 unless you're one of the minority who get a 'golden' chip by fluke.
I'm not saying the 8300 isn't a better chip, it is but not at stock speeds irregardless of the 2 extra threads.
& with that board a 6 core is far more suited for overclocking than an 8 core.
As weberdarren97 mentions above if you have an nvidia 7** series or newer offloading to the nvenc encoder removes a huge amount of CPU overhead for streaming.
All the above is going to be dependant on the main titles you're streaming aswell ?