Some properties give suite upgrades to me as a Diamond that didn't give them to me when I was Gold. And I've spoken with several managers who mentioned that they try to upgrade Diamonds first among guests arriving that day. At other properties I don't see a difference.
The ability to get award rooms is the most reliable difference, I'd say. When I've wanted an award stay in a sold out property, the Diamond line hasn't hesitated to arrange for me to get the room I wanted.