Which Ryzen - depends on your needs. If you want the CPU to last between 3-5 years for gaming, the Ryzen 5 1600 is an excellent choice. For more than 5 years in gaming, or if you are a content creator(video editing, rendering, animation, etc), then the Ryzen 7 1700 will be better. Either way, for 60 FPS gaming, don't go with the i7, at least not the 7th gen ones.
What's the difference - more cores. An i7 7700 is 4 core, 8 thread, and costs more than a 6-core 12-thread Ryzen 5 1600. Also, Ryzen processors are overclockable on cheaper B350 boards, while you can't overclock the 7700 at all. The Ryzen processors also come with very nice coolers, and on stock cooling they will always overclock to at least 3.7 GHz, if not more. On more cooling, you can hit 3.8-4.0 GHz as well.
The difference, in terms of what matters, is that a Ryzen processor will last longer in games, while an i7 will not. Games have started using more than 4 cores(or they soon will, depends on how you want to look at it), which is why a current gen i7 is not considered future-proof anymore. Now, Intel will be launching their next gen 8th gen CPUs soon, and they are rumored to have 6 cores. But you'll have to wait for them to launch. For now, it's just Ryzen.
Another major difference is high refresh rate gaming. AMD has always lagged a bit in that field. Intel CPUs have higher IPC and clock speed, which helps them in delivering at high refresh rates, ie, 144+ Hz. Ryzen processors, on the other hand, lag behind in such high FPS due to lower clock speed. Since you'll probably game at 60 FPS, the Ryzen is better, since your monitor won't be able to display more than 60 FPS(or so I'm assuming).
One final difference is upgradability. Ryzen processors are on the AM4 socket, which was introduced only in March this year. On the other hand, 7th gen Intel is on LGA 1151, which is now an older socket and you won't be able to upgrade to the 8th gen CPUs if you buy the 7th gen motherboards. On the other hand, AMD has committed to the AM4 socket till at least 2020, which means that you can buy the next generation on AMD processors and keep the same motherboard for using them.