How does Diablo still exist?

Solution 1:

There is the scene where the black soulstone is completed in Act II. It is explicitly stated by Adria and accompanying adventurers that that is what she has been doing all the years of Leah's life. Going around and marking the souls of the three previous prime evils so that when the black soutstone was completely they would be sucked into it.

So the current in-game explanation is that the souls were never destroyed, they were then sucked up by the black soulstone when it was completed.

Solution 2:

If you want spoiler-free lore background, I can only recommend this video of the Blizzcon 2011 Diablo 3 Lore Panel. To put it in a nutshell, you can't really kill the great evils, only their manifestation in Sanctuary.

The soulstones where created to capture the essence of the demons, and bind them, since they were almost indestructible. Chris Metzen said in an interview that the whole idea of the soulstone was influenced by the "exorcist" idea of sacrificing yourself as a vessel to capture a demon. That is why the hero from Diablo 1 becomes the dark wanderer in Diablo 2.