As I understand it (but I have no inside info):
-the PR number or UCI on the front is unchanging and will stay with a PR/citizen (or even one-time TRV holder) for life. That's how you identify yourself, basically. It's irrelevant whether the card is 'valid' or whatever else - it's used to identify you (or more accurately perhaps, your file(s) in the system). Hence unique client identifier. Largely speaking it's the most relevant.
-I believe the number on the back is the physical card identifier. You may get asked to enter it in certain circumstances to ensure that you have a current, valid, physical card, including one in the database. Changed any time they issue a new one (and flag old ones as expired or stolen or lost, for example).
I assume that this latter number may also have some security features/mathematical formulae behind it that allows it to be identified as (potentially) a real card, i.e. a random string of letters and numbers can be identified as fake, and/or links it somehow to the holder of the card (checksums of some of the info on the front encoded in that number). Note, I don't know anything about this, but it's pretty simple stuff to do - it would allow check-in personnel to key in or even read over a phone small subsets of data and get some kind of 'seems legit' or 'check again' response. Or in simple terms, somewhat like a credit card number can be identified as not real with a simple algorithm. Not the only security feature of course but one that doesn't require physical examination.