the 8350 would benefit streaming, but would likely be a downgrade in most other areas. the 8350 is good at multi-threading due to its 8 cores (so streaming games wouldn't impact performance quite as much as your current cpu), but the cores are very weak individually, making it suck for anything that uses less than 8 cores (which most things do).
to answer your question about if the 8350 isnt better than the i7 and why, this is because the IPC (instructions per cycle) is low on an 8350, about half of that of a modern intel cpu. think of it like a wheel, where 1 rotation = 1 cycle, the rotations per second = the cpu frequency (GHz), and the size of the wheel (the torque) = the instructions per cycle (IPC).
let's say we have 2 wheels, both spin at the same speed 4000 rotations per second (4GHz), but one of the wheels is twice the size as the other. the bigger wheel is going to travel a lot further than the smaller one. this is because the bigger wheel travels a much longer distance in a single rotation, which is essentially the same reason an i5/i7 is much faster than an 8350 most of the time; the intel cpu accomplishes a lot more in a single clock cycle (1Hz essentially) than the AMD. it's a pretty basic analogy (a CPU is obviously WAY more complex than a fucking wheel), but it is pretty much the reason the 8350 is so much worse than an i5/i7 most of the time (there are some things an 8350 could beat intel at, which goes in to reasons beyond the simplicity of this analogy).
you would also have to buy a new motherboard if you wanted to get an 8350, so i would just say put that money towards an i7 4790k. the 8350 is priced nicely but most people (me included) regret buying them in the end after realising they're just way behind intel. like i say though, you'd also be spending almost as much on the 8350 on a motherboard to go with it, which is going to cost something similar to the i7 (which wont need a new motherboard).
and about the DDR5 VRAM, its just like normal system RAM; more of it would give you a huge boost in performance if you were to run out of it; if you weren't running out of normal system RAM in your computer, it would be pointless buying more. 1GB is still enough for the majority of modern games (big titles like GTA 5 or witcher 3 would certainly need more though) and definitely enough for older games including tf2. in your current situation, you don't need to worry about VRAM or a new gpu.
in short; for streaming tf2, an i7 would be the best option. the 8350 is also pretty good at streaming, but shit at everything else (you would probably get a lower fps on tf2 when not streaming than your current 4570) and you would need a new motherboard anyway. an i7 still beats an 8350 at streaming, just not with as much of a difference as everything else.
your current gpu is enough for tf2 and wouldnt benefit streaming, but it is a pretty old mid-range card. if you were looking to play other, more graphically-demanding and newer games, you should definitely put the money towards a new gpu instead, as your i5 is good enough for playing modern games when not streaming.
hope this helps!