"It may rain on the weekend."
会 makes it more definite, and in this case more definite that it might rain. If that sounds contradictory, then consider what I write below. It's really about degrees of prediction.
会 is a verb that indicates that a certain event is something that will definitely occur in the future. Therefore 会 can mean something 'will' happen or can happen.
If you notice in some of the other answers they only use 会 when they are talking about a weather forecast, so it is much more certain that a particular weather event - rain, overcast, snow etc. - will happen. There's one answer here for example that says "it will be windy this afternoon" where they only use 会, making it a definite prediction.
The moment you put in 可能 then it is maybe, or might, and some doubt is added to a predicted outcome.
So I would say when you add 会 to可能 it makes the situation like it is 'fairly likely' and no longer maybe.