I had the opportunity to use both stones (but not recently), which offer similarities and differences depending on the desired objectives. The main common points: both stones are relatively soft, but it's their ability to achieve a "quick" kasumi finish quite easily (very contrasted of course, without the level of subtlety one would get with jnats) that characterizes them (the King800 is known for it). Regarding some obvious differences: the JNS is a soaker, more expensive, but also much bigger (1400 gr. after one year), and dealing with slurry management is more pleasant than with the King, which is a splash and go, also often associated with a "mud bath".
However, if money is a concern and for versatile use, the King is a good value, in terms of performance and quality-price ratio. That being said, I have a preference for the JNS, which I find faster, especially in the context of polishing (if that's something you would consider), as it provides very consistent results, effectively helps to blur the scratch pattern of coarser stones (such as the JNS 300, for example), and above all, I appreciate the schalky feedback it offers, a recurring characteristic of JNS stones. But this JNS800 also has also some disadvantages from my experience: As it is quite "soft," it tends to wear quickly if too much pressure is applied. Therefore, it's important to use the entire surface and consider flattening it with an Atoma, for example. Additionally, it is very thirsty, and even though it is a soaker, it will need to be left submerged for a long time to fully benefit from its qualities. More experienced polishers will likely criticize it for hiding defects, low spots rather than correcting them.