The story of D2 was awesome, but my experience of the story was much less clear while I was playing the game. I actually went to the Diablo wiki and filled myself in to really appreciate the story. The clues are there for you to pick up if you are attentive while playing the game, but I really didn't understand the the story very well until my hell playthrough. In D3, I felt I had a very good handle on what the story of the game was, but the story itself, especially act 3 and 4 left something to be desired. It just wasn't that compelling.
I was left wanting something from Mephisto and Baal. If you didn't play D2, you wouldn't really know who they were unless you listened carefully to the writings on the lords of hell by Cain in act 2. They are supposed to be equal to Diablo in their power and evil...and yet nothing about them or influence from them.
The butcher could have used some hyping up in the buildup to the final boss of act 1. In all the other acts, you knew who you were gunning for, but the Butcher kinda comes out of nowhere with a pitiful introduction from Mahgda. How did the butcher come back after he was defeated in Diablo 1? I wanna know. Dont need anything super detailed, but at least some mention of it would be cool.
Azmodan, greatest battle commander in the burning hells...reveals all his plans to the hero and how to stop him. Not much of a tactician. Other people have touched on this, and I just wanted to add my tally to that vote.
Leah becomes Diablo...I kinda felt the same way about her "death" as I did Cain's. It needed something more.
In D2 act 4, you defeat Izual and supposedly free his soul from its corrupted demon form, and even have a conversation with the angel, who spurs the hero on to face diablo...and then all of a sudden he's a demon again under Diablo's command. Why? I need some story here behind that.
I'm hoping Pindle comes back as a miniboss in act 5, or at least a rare spawn...it seems like it would be a good fit with the whole army of the dead theme that act 5 looks like it will be following. Just for a bit of D2 nostalgia.