Full Idea
Weak supervenience says there is no possible world that contains individuals that are B-indiscernible but A-discernible. Strong supervenience entails the same even if they are in different possible worlds.
Gist of Idea
Weak supervenience is in one world, strong supervenience in all possible worlds
Source
Karen Bennett (Supervenience [2011], §4.1)
Book Reference
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.14
A Reaction
In other words (I presume), in simple language, the weak version says they happen supervene, the strong version says they have to supervene.