The i3 is newer, high clock rate which works better for lowly threaded apps.
If you are streaming though, you will want more performance for threaded apps, as you game will use one or 2 typically, and the streaming app will use the others(hopfully).
The i3 is a hyperthreaded dual core, so it will run 4 threads, but the performance on 2 of them is less.
The i5 is 4 real cores, quad core, but not hyperthreaded so also 4 threads, but all 4 are legit full performance.
But the i5 is an older platform which is good/bad. Good for prices on current parts, bad for prices in the future as older parts get more expensive as they leave production and stock runs out.
The i5 has higher general performance in the end I think for your use case.