I respectfully disagree. In general, Cartier diamonds are considered "better" than Tiffany's - assuming we are doing an exact apples-to-apples vsII-vs-vsII comparison. (By better, I mean as viewed by most jewelers or jewelry collectors. I used to work for an auction house, so I share a general industry view.)
You will be hard pressed to find any jeweler who will rate Tiffany diamonds higher than Cartier (again assuming all things are equal.)
It is true, however, that Cartier does give a discount between 10-20% off sticker price, depending on the store and manager.
Musicscrip, best of luck finding the diamond ring of your dreams!
I believe one of the reasons why Cartier is more expensive for engagement rings is because they are French based? Tiffany is more reasonable here in the States but I know that they are even more expensive abroad.
Vintage/Estate Cartier probably goes for more at auction because it's reputable, it's French and outposts to buy authentic Cartier are few and far between (versus most cities having a Tiffany and larger markets having several).
Though really, all of the big designer names aren't really competing or in a hierarchy in my eyes. They all cater to different tastes... I don't think Tiffany (more modern), Harry Winston (glllaaaaaamazon!), Van Cleef (super whimsical), Cartier (romantic/antique), Niessing (ultra-modern)... I think if you have a piece of two displayed in the special gems room at the Smithsonian, you're pretty top of your field.