1) The Core i3 will run cooler and be faster in single threaded operations.
2) The Core i3 is also more efficient and uses a newer manufacturing process.
3) Intel's upgrade path is much better than AMD's. For example, if you buy a i3 now, you can upgrade to a i5 or i7 at a later time.
Ideally, a i5 is perfect for gaming. But I would recommend the i3 for this build.
*you will need an Intel motherboard too if you go with an Intel chip*