I think its generally understood that most artificial sweeteners are seen as sugar by the body, but don't actually get turned into sugar in the bloodstream. So it makes sense that it might contribute to diabetes (your body digests it like sugar), but not aging (which is accelerated by sugar in the bloodstream). I believe the whole point of coke 0 is that it doesn't use an artificial sweetener? But don't quote me on that last sentence.
Coke Zero for the US market still uses aspartame, the same sweetener used in Diet Coke.
Coke Zero is a more modern formulation which is supposed to taste more like "classic" Coke than Diet, which has its own distinctive "artificial" flavor which some customers have become accustomed to. It's also marketed very squarely towards young men: during the recent "Share a Coke with X" campaign, there was even a "Share a Coke Zero with a BRO" can, which made me laugh a bit.
As the others have said, Coke Zero contains aspartame. However, it also contains acesulfame potassium which is another sweetener, so it may have a little less aspartame than Diet Coke. (The effect of multiple sweeteners is generally greater than an equivalent larger dose of one... or so I have... read. Maybe. In something somewhere. No promises.)
They seem to have gone back and forth, and more it varies by country.
My experience in Sweden was that first they with big fanfare changed Coke Light (what they call Diet coke outside of English-speaking countries) to Sucralose, then introduced Coke Zero, which used Sucralose and silently changed Coke Light back to Aspartame, and then at some point silently also changed Coke Zero to Aspartame.
There's a version of Diet Coke sweetened with Splenda, but Coke Zero is sweetened with aspartame just like regular Diet Coke. The Splenda Diet Coke isn't bad, but it does taste a little different than normal Diet Coke or Coke Zero. Look for the Diet Coke cans with a yellow label to find the Splenda version.