Author's Note: This is post-Eclipse, and references some things mentioned on the Twilight Lexicon but not in the series.
Esme's reason is probably the most compelling. "You're going to marry her soon. Husbands and wives are open with each other. They don't keep things hidden. They don't try to control what each other knows and doesn't know, does and doesn't do." She looks at him hard, and Edward knows even without hearing her thoughts that she isn't speaking about her life as a vampire, but as her life as a human, and he wants to tell Bella because he doesn't want to be Charles Evenson.
Carlisle's reason isn't quite so powerful. "You always assume that she's going to react badly to what you have to say, but your attempts to go around that have caused more trouble in the past. If you had been honest about leaving, she wouldn't have been so upset. Well, if you're honest about this, she'll be less likely to get upset. Bella always finds out," he adds, and even though he knows that she wouldn't guess this on her own, Edward wants to tell Bella because even though he doesn't have those feelings for Carlisle anymore, he respects him more than anyone in the world.
Alice's reason is mildly scary. "I've seen what could happen if you don't tell her," she says to him one day. "She will find out, Edward- if not from you, from one of us, or she'll figure it out on her own. It won't be pretty. I saw her, and she was very offended, probably because you didn't trust her to still love you after she knew. Or because you lied to her, but that's less likely. She knows that a hundred years is a long time. She probably didn't believe you when you said she was the only one you ever felt anything for." Alice changes her mood suddenly, and she's back to getting him to pick a wedding cake, and Edward wants to tell Bella because he knows he'd do a better job breaking the news than Alice would.
There are other reasons, too. "She understands what it's like to have her life saved by someone," Jasper points out. "That was years ago! Just tell her you were still a kid, and you were confused," Emmett jokes. Rosalie is the only one who disagrees. "It'll hurt her pride," she insists.
But Edward has reasons for staying quiet that surpass them all. He doesn't want to tell Bella because he's afraid that she might reject him. He doesn't want to tell Bella because it's in the past, and he loves her now, and he doesn't want to think about loving anyone else. He doesn't want to tell Bella because it would wreck the perfect image of the Cullen family. He doesn't want her to know that he hadn't always welcomed the role of Carlisle's son with open arms, and that he had hated Esme for things that were beyond her control, and that his experimentation with a traditional vampire life was the culmination of years of hidden anger. He doesn't want to tell Bella because once she knows everything about him, and understands him completely, she'll be a Cullen, and he's not sure he's ready for that.