There are a lot of differences.
Price difference: R5 1600 build will be ~$120-$150 cheaper.
Performance difference: i7 can run faster fps if graphic card is extremely fast, or resolution is very low. Fast cores ensure games like CSGO run at best fps. R5 can run games well, especially if resolution is high, and do extra tasks at the same time. Extra cores ensure games like Battlefield1 in multiplayer can run very well.
Both can be overclocked by ~10-15%, R5 comes with a box cooler that can do that.
R5 is faster in productivity if you care, and should be about as fast in streaming.
R5 is on a new AM4 platform, so motherboards will get new processors for a few more years. i7 is on a platform that will change this/next year.
In the very long run- their performance in games should equalize, but for a while i7 is faster and costlier platform for games, although everyone is reporting game feel on R5 about as good, if not smoother. General wise choice would be to save money and get R5 1600, and put whatever is left into GPU and monitor.