Is the sentence, "There is nothing common between you and me." wrong grammatically? If yes, why? Should it be, "There is nothing in common between you and me."?
Both sentences are grammatical, but they can mean different things.
Which you should say depends entirely on what you're trying to express.
1. There is nothing common between you and me.
This could be interpreted to mean there is nothing [that is considered] ordinary [located] between you and me. For instance, what's between us on the table is not an everyday hamburger but a rare and expensive five-course meal.
2. There is nothing in common between you and me.
This means that the two of us don't share anything that is the same. For instance, we will never agree on anything because we don't share any of the same opinions.
From Merriam-Webster's definition of common, there is an entry for the phrase in common specifically:
: shared together
// has a lot in common with his neighbors
The first could be interpreted to mean the second, but if you're talking about sharing something, it's more likely to be understood if you add the preposition in to the sentence.