If $K$ is a pretopology on a category $\mathcal{C}$ and $J$ the topology it induces, are the Grothendieck toposes $\text{Sh}(\mathcal{C},K)$ and $\text{Sh}(\mathcal{C},J)$ the same in general?
As I say in the title, I would like to know if one can define sites as categories endowed with pretopologies instead of topologies, in the sense that the resulting toposes of sheaves do not become structurally different. If not then I would like to know what is lost.