Arguments are given which point out that the notion of "conformal equivalence" -- as it has been used in the bibliography on the issue to date -- is lacking in physical and mathematical content. Unless the meaning of conformal transformations for the underlying spacetime geometry is discussed on a same footing as it is done for the relevant equations of the given gravity theory, the issue is no more than a semantic one. To put the discussion on concrete grounds a principle of conformal equivalence is formulated and an alternative geometric interpretation of the conformal transformations is given. This allows demonstrating that, contrary to previous claims in the literature, vacuum Brans-Dicke (BD) theory of gravity ($\omega\neq -3/2$) is not conformally invariant and, consequently, the different conformal frames in which the theory can be formulated are just different -- conformally related -- representations of a same phenomenon. These can not be compared: either all of them are physically correct, or none is it. To illustrate our arguments, we explore a modification of BD theory of gravitation which is indeed conformally invariant. The different points of view existing in the bibliography about the (non)equivalence of the Jordan's and Einstein's conformal frames of Brans-Dicke theory are critically analyzed on the light of the formulated principle of conformal equivalence and of the alternative geometric interpretation of the conformal transformations given here. Our results can be straightforwardly applied to scalar-tensor theories in general.