What's the difference between DDR3 PC3-1300, PC3-1600, PC3-12800, etc

PC3-12800 (12800 MB/sec) is the same thing as DDR3 1600MHz (1600 MHz data transfer rate) and it runs at 800 MHz DDR clock. Similarly for others, just divide/multiply by 8.

Any memory that has the correct voltage and capacity and is same or faster than what is supported by the board should work in the board. The memory will run at 1333 MHz (PC3-10700), unless you enable overclocking (XMP or other) on the board because the CPU likely doesn't support 1600 MHz within its specification.

So in your case, there should be no issue replacing your RAMs with the new pair. I say should as there can always be incompatibilies beyond what is covered by specification.