Scientific truth is not the same thing as religious truth. There can be true statements in each system which contradict each other. Whether or not it is meaningful to compare truths arrived at by different methods is debatable.
Science is rooted in empiricism; truth is determined by observation. It's well suited to producing truths about things which can be isolated in an experiment. Things like economies, law/morality, and mathematics are more or less intractable for science, and it cannot make many statements about supernatural beliefs. There is a branch of philosophy which regards the unproven or unprovable as false, but it's a minority viewpoint.
Religion is (perhaps surprisingly) fairly rational, as in rationalism. However, fundamentally its truth is rooted in received wisdom and not subject to test by observation (and to some degree not to reason either). Where religion makes statements about the observable world, it can and often does conflict with empirical truth. Sometimes these things are called miracles.
It is not true that the realms of science, religion, and logic do not overlap. Religion does not always confine itself to the intangible. Scientific theories are always dancing on the limits of observation. Personally, I don't think it's useful to try to reconcile different ways of determining truth, and I think you and John Paul II are with me on this one. I'm not aware of any sensible way to establish a value relation among them; each is necessary in its own domain.
That said, while in my moments of intellectual honesty I am agnostic, believing strongly in something contradicted by empirical truth is not usually something I can take very seriously.