The 6800k is laughably underclocked at stock just like the 5820k was.
I run my 5820k at ~4.2GHz and stock on it is 3.3GHz.
That said, the 7700k also OC's a fair bit as well, so it will still have the stronger cores, but if you are streaming those two extra cores can be nice to have.
If you dont need super high quality streaming however you can use the 7700k's iGPU for encoding your video for streaming, whereas the 6800k doesn't have an iGPU and you would need to use CPU encoding, or have another GPU available for GPU encoding.
For simplicity sake, the 7700k is probably the better option.