In terms of overall gaming performance, the AMD Ryzen R5 1600 is massively better than the AMD Athlon X4 880K when it comes to running the latest games. This also means it will be less likely to bottleneck more powerful GPUs, allowing them to achieve more of their gaming performance potential.

The Ryzen R5 1600 was released over a year more recently than the Athlon X4 880K, and so the Ryzen R5 1600 is likely to have better levels of support, and will be more optimized for running the latest games.

Both CPUs exhibit very powerful performance, so it probably isn't worth upgrading from one to the other, as both are capable of running even the most demanding games at the highest settings (assuming they are accompanied by equivalently powerful GPUs).

The Ryzen R5 1600 has 2 more cores than the Athlon X4 880K. 6 cores is probably excessive if you mean to just run the latest games, as games are not yet able to harness this many cores. The cores in the Athlon X4 880K is more than enough for gaming purposes.

The Ryzen R5 1600 has 8 more threads than the Athlon X4 880K. The Athlon X4 880K has one thread per physical core, whereas the Ryzen R5 1600 uses hyperthreading and has 2 logical threads per physical core.

Multiple threads are useful for improving the performance of multi-threaded applications. Additional cores and their accompanying thread will always be beneficial for multi-threaded applications. Hyperthreading will be beneficial for applications optimized for it, but it may slow others down. For games, the number of threads is largely irrelevant, as long as you have at least 2 cores (preferably 4), and hyperthreading can sometimes even hit performance.

More important for gaming than the number of cores and threads is the clock rate. Problematically, unless the two CPUs are from the same family, this can only serve as a general guide and nothing like an exact comparison, because the clock cycles per instruction (CPI) will vary so much.

The Ryzen R5 1600 and Athlon X4 880K are not from the same family of CPUs, so their clock speeds are by no means directly comparable. Bear in mind, then, that while the Athlon X4 880K has a 0.8 GHz faster frequency, this is not always an indicator that it will be superior in performance, despite frequency being crucial when trying to avoid GPU bottlenecking. In this case, however, the difference is probably a good indicator that the Ryzen R5 1600 is superior.

Aside from the clock rate, the next-most important CPU features for PC game performance are L2 and L3 cache size. Faster than RAM, the more cache available, the more data that can be stored for lightning-fast retrieval. L1 Cache is not usually an issue anymore for gaming, with most high-end CPUs eking out about the same L1 performance, and L2 is more important than L3 - but L3 is still important if you want to reach the highest levels of performance. Bear in mind that although it is better to have a larger cache, the larger it is, the higher the latency, so a balance has to be struck.

The Ryzen R5 1600 and the Athlon X4 880K have the same L2 cache size, but the Athlon X4 880K does not appear to have an L3 cache, so the Ryzen R5 1600 definitely wins out in this area.

The maximum Thermal Design Power is the power in Watts that the CPU will consume in the worst case scenario. The lithography is the semiconductor manufacturing technology being used to create the CPU - the smaller this is, the more transistors that can be fit into the CPU, and the closer the connections. For both the lithography and the TDP, it is the lower the better, because a lower number means a lower amount of power is necessary to run the CPU, and consequently a lower amount of heat is produced.

The Ryzen R5 1600 has a 30 Watt lower Maximum TDP than the Athlon X4 880K, and was created with a 14 nm smaller manufacturing technology. What this means is the Ryzen R5 1600 will consume significantly less power and consequently produce less heat, enabling more prolonged computational tasks with fewer adverse effects. This will lower your yearly electricity bill significantly, as well as prevent you from having to invest in extra cooling mechanisms (unless you overclock).