Note that in English the uses of some connectives appear to be truth functional, whilst the uses of others clearly are not.
What (a) claims is true if both John and Mary ate steak, whilst this is not sufficient for the truth of (b).
Clearly, the importance of truth-functional connectives is that we are guaranteed to be able to compute the truth value of a complex sentence by taking it apart according to the syntactic rules and computing the truth values of the parts.
This process, known as ``truth recursion", terminates when we reach an atomic sentence whose truth or falsity is established by looking in the world.