Equivalent formulae with different CNF I was not able to find or come up with two formulae which are equivalent but have different CNF. All my ideas reduce to the same formula after applying transformations. The requirements are the following:

Give an example which shows that proving equivalence of two formulas by bringing them into conjunctive normal form is incomplete. By incompleteness, we mean that there are examples of formulas which are equivalent but their conjunctive normal form is not the same.
It depends on the algorithm for obtaining CNF of a formula.
A simple truth table-based method will always produce syntactically the same CNFs for equivalent formulas, since it in a sense just gives you the rows of the truth table for which the formula evaluates to false.

A trivial example of equivalent formulas, which are both in CNF:

$$A \wedge B$$
$$(A \vee B) \wedge (A \vee \neg B) \wedge (\neg A \vee B).$$

Non-uniqueness of CNF means that one can build an algorithm, which returns essentially different CNFs for equivalent formulas. For example, as a first step the conversion algorithm can check if a formula already in CNF and return it immediately.