The Ryzen 3 2200G is the replacement for the first gen Ryzen entire Ryzen 3 line up. It is clocked higher and has integrated graphics that make for a good budget gaming PC. It has 4 cores but no SMT (AMD's version of hyperthreading).
The Ryzen 5 1500X is a 4 core chip with SMT, giving it 8 threads. It is clocked faster too and reaches a higher turbo speed. It does not have integrated graphics. However, it is slightly less efficient than the 2200G.
As long as you have a reasonably powerful video card, go with the 1500X. It will offer similar performance in most games, but in any that respond well to more cores it will offer more performance.
Hope that helps.