At least for me, when I get upgrade offers (from PRG to Platinum) they will let me keep my same card number, etc. Which is nice from an auto paying bills perspective.
It's definitely worth calling. Tell them what offer you're seeing for a new card, and ask if there's reason not to open a new one and cancel the old one. Best case, they offer to match (or come close) to the new card offer. Worst case they give you some reasons you'd already read or thought up yourself.