In theory the 8 cores/threads of the fx 8xxx should be superior but reality is a bit different. For instance adobe premier pro cc, h.264 encoding will use up to 10 cores efficiently.
https://www.pugetsystems.com/labs/articles/Adobe-Premiere-Pro-CC-Multi-Core-Performance-698/
Yet in TH's tests using premier pro to encode h.264 in premiere pro cc, the 4670k i5 4c/4t took only 1 second more than the fx 8350 and the i5 4690 encoded 4 seconds faster than the fx 8350.
http://www.tomshardware.com/charts/cpu-charts-2015/-31-Adobe-Premiere-Pro-CC,3722.html
That's just one test but a situation where it's confirmed the software can use far more than 4 cores effectively and yet the i5 with half the cores outperformed the 8core fx. Simply stating the fx is better because it has more cores/threads and the application is 'heavily threaded' isn't always true in the real world.
In streaming, hard to say. It's hard to give a definitive benchmark result because of all the variables. Are you using a single drive, a separate drive to encode to from the source video, are you encoding while gaming, the speed of the drive(s), the speed of the upload connection, what program are you using to encode with, what settings, how many fps and at what resolution. 720p at 30fps, 1080p at 30fps, 1080p at 60fps? There are endless combinations and it would be extremely difficult to give a direct answer because of that.
Personally I would opt for the i5. If you need something more your system would already be set up to drop in an i7. If you go with amd, you're stuck with no place to go. There's no drop in replacement for the fx 8xxx that's more powerful, it's already the best of what amd offers. In order to upgrade from the fx you'll be looking at a new cpu, new motherboard, new ram most likely (ddr4 for either skylake or amd zen) and a reinstall of windows.