One of my hero's, the late Chuck Meadows explained it to me in very simple terms; 347 is really just a stabilized 308... so, based on that simple analogy, you could compare 347 to 308 very nicely. 347 withstands higher temperatures better than 308 but for welding purposes, you wouldn't know the difference. 316 is a higher alloy and has it's place. What's the application? I believe generally speaking, you'll probably pay more for the 316 and if it's required, by all means use it, if not, don't worry about the 347, you'll like it.
I'm sure js55 could and maybe will provide a more in-depth discussion on the technical aspects but above is somewhat of a "users" recommendation.