The biggest difference between Diet Coke and Coke Zero are the base formulas and the sweeteners.
In short, Diet Coke is based on the "New Coke" formula, but with artificial sweeteners (aspartame and acesulfame potassium). In contrast, Coke Zero is based on the "Coke Classic" formula, also with artificial sweeteners (aspartame). Thus, when people say that Zero tastes more like Coke, it's because it actually is. Diet Coke isn't a diet version of what you typically assume is Coke.
I wrote a little about the contrast via these posts on BevReview.com:
Branding of Diet Coke Review: Coca-Cola Zero Packaging Change: Coca-Cola Zero
Incidentally,
Diet Coke Plus also uses aspartame and Ace-K, which I sort of felt at the time was a way to try the new formula on the Diet Coke audience and see what they thought. It's much sweeter. The only other Diet Coke-branded drink with a different sweetener blend is the obviously-named
Diet Coke Sweetened with Splenda 