If looking at the hierarchy chart, most of the FX-83xx CPUs are in the second tier.
Assuming Zen is equal to the current top tier Intel CPUs, then the only major benefit is power efficiency, and some performance gain.
If they wind up notably better and/or have more overclocking headroom... well, that's different story.
Honestly, though, until the product and the full tests/reviews come out, it's all very speculative. There are no benchmarks because no Zen CPUs are out yet. I think they were talking beginning of March.
None of my games stress my current system. The only reason I'm considering upgrading currently is because my son's PC is very budget-oriented. I'd give him my current system, and get a new one for myself.