I am trying to give two differences between 再 and 又 for you.
Originally, 再 means 'second time', 又 means 'second time', or 'third time', or even more. But now, 再 sometimes also means repeatedly. In this sense, the two words,再，又, are replaceable.
再 emphasizes the two concepts (verbs/nouns) must be the same nature, while 又 is not limited.
The more important thing is to know some examples.
Usually, the 再 in the following examples can not be replaced by 又.
再見。 See you a second time, so it means 'see you', 'see you later', 'see you again', or 'Auf Wiedersehen' in German.
再會。 This sentence has the same meaning as 再見.
一而再，再而三 Do the same thing once, twice, and three times, so it means 'again and again.
再三 Do the same thing twice and three times, i.e. it means 'again and again' too.
The 又 in the following examples can not be replaced by 再.
[既...又...] 襄兒既聰明,又可愛。 Xianger is not only bright but also cute.
[又...又...] 又驚又喜；又快又好 Shocked and happy； fast and good
[又名] 中國，又名海内，諸夏。 China is also known as '海内' or '諸夏'.
牛頓來了，又走了。 Newton came and then left.
For your question, 再會下雨 is indeed a wrong expression. Let me try to make is clear.
再會下雨 means "again 'could rain' ". But once today is coming, 'could rain' will become 'rain' or 'not rain'. So the same thing 'could rain' will not happen again tomorrow. On the other hand, 會再下雨 means 'could rain again'. There is no problem in this expression.
In the above sentence, 又會下雨 is right, because 下雨 and 會下雨 can be joined by 又。
PS: Thanks for your interesting question. A piece of advice from me: I think you'd better learn from examples, it is helpful.