When is a connective truth functional?
Short answer : when it is defined by a truth table.
Classical propositional logic is a truth-functional logic in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. On the contrary, modal logic is non-truth-functional.
See an example in Truth Functionality and non-Truth Functional Connectives, comparing :
Agnes will attend law school and so will Bob,
where the truth-value of the compound sentence depends only on the truth-value of the two atomic sentences, with :
Agnes will attend law school and then she will make millions,
where the "and then" connective express a time-dependency between the two atomic sentences.
For different examples, see 6.3.1 Indicative and Counterfactual Conditionals (page 110) of Smith's book.
An example (motivated by your previous question) dealing with the concept of "internal structure" of a statement will be the following.
"Jim is a bachelor and Jim (the same Jim) is married"
is not a contradiction in propositional logic, because the sentence has the logical form B ∧ M, and this formula is not a contradiction.
In order to discover the contradicition, we need a deeper level of analysis that consider also the semantics of the expressions "is a bachelor" and "is married", in addition to the logical connective "and".
This level of analysis will be available with predicate logic where we can analyze the atomic sentences with a subject-predicate logical form :
Bachelor(Jim) and Married(Jim).
In this case, using the axiom :
Bachelor(x) iff not Married(x),
we may derive the contradiction not expressible in propositional logic.