"So what happened between yesterday and today" is better here. Between is a preposition used in relation to two or more objects. Whereas "in between" is an adverb, i.e it relates to the verb in the sentence. Usually, "in between" is used when the objects occur prior to the verb, such as: "There was yesterday and today, so tell me what happened in-between?"

Between and In-between technically are different in their usage/form of grammar. "Between" is more of a connection word for objects, whereas "in-between" will be more of a location within objects A and B. Oxford online dictionary has lots of example uses if you wanted to know more about it grammatically. That said,informally, here in America at least, we'd say "what happened between yesterday and today?" if we were implying there was some sort of situation change between the two days.

I think "between" sounds better there.