The bilingual advantage is a theory (I guess, actually a hypothesis) that states that the brains of bilingual children develop differently. Specifically, it suggests that multilingual children are better at task-switching (going from one kind of thinking activity to another) and executive function (the ability to manage higher-order thinking skills).
A researcher analyzed the working drafts of 13 years of conference presentations and finds that this advantage might be over-stated. I read about it in the New Yorker.
I have always been of the opinion that the advantage of speaking another language is your ability to speak another language. I'm not embarassed to say that I've talked about bilingual advantage (the neurological theory) to sell Spanish classes and other programs to decision-makers. It seems I'll have to take this new research under consideration.