Too often, games seem forced to have the plot and the gameplay largely separated, making it seem like you're either playing a game that is occasionally interrupted by a story, OR you're watching a movie that you can only see in its entirety if you play the game.
With all that said, I really liked how Prince of Persia played some of the character-developing dialogue during the gameplay. The fact that it was typically monologuing still added depth and believability to the character. I find it hard to believe that a person can run around with sealed lips and a blank mind ALL the time. I think a game that takes what PoP has done and turns it up to 11 would be a game worth playing. Example: When learning about a target in Assassin's Creed, instead of a bit of text telling you about it, how much cooler would it have been to hear the guy's inner thoughts going "This man's ego is bigger than his head if he lets his guard down just so he can make a speech." It's those little touches that makes a game stand above the rest.
WOW, where the hell did that tangent go? Okay, tl'dr, Prince of Persia had good storytelling because character-development happened while you played. Also, it eventually becomes a story of ultimate redemption, knowing your priorities, saving the girl, yada yada blubbedy blah.
Also, Professor Layton tells a good story... when you're not being interrupted by puzzles every 5 goddam seconds. "Say, citizen, I'm wondering if you've seen this man lately?" "No, BUT WHY DON'T YOU SOLVE THIS AWESOME PUZZLE I COOKED UP?!"