TITLE: difference between the small and big étale/flat/... site
QUESTION [9 upvotes]: What is the difference between the small and the big étale (or flat or syntomic or ...) site? How does the cohomology vary? When should I use which one? Up to now, I have always used the small sites.

REPLY [7 votes]: You can always compute the cohomology as the limit over hypercoverings of the final object. But the hypercoverings of the final object are the same in both categories, so it seems that the cohomology is the same (i.e. the cohomology of a sheaf on the big etale site maps isomorphically to the cohomology of its restriction to the small etale site).
Big sites are useful e.g. when you want to consider schemes as etale (or fppf or ...) sheaves. If $S$ is a scheme, then an $S$-scheme $X$
 gives and is determined by its associated sheaf ${\rm Hom}_S(−,X)$
 on the big etale site of $S$. So you can define e.g. the Picard scheme by defining the associated sheaf first and then asking whether it is representable. This utterly fails for the small etale site though: take $X\to S$ a closed immersion, then the restriction of ${\rm Hom}_S(−,X)$
 to the small etale site will usually be the empty sheaf.