Why does Montresor wait fifty years to tell his story? Would the story differ if told the morning after?

If Montresor had confessed the same day, he might have been caught.  He is very concerned that he not be caught because that is a major point of pride for him.  I also think he would have been too tired to tell it the next day after staying up all night bricking up the wall!

Why does Montresor wait fifty years to tell his story? Would the story differ if told the morning after?

All true.  And don't forget one of the criterion for revenge set by Montressor himself:

I must not only punish but punish with impunity. A wrong is unredressed when retribution overtakes its redresser.

He may be crazy, but Montressor knows he has to wait to tell the story until much later in order to achieve his full measure of vengeance.

Why does Montresor wait fifty years to tell his story? Would the story differ if told the morning after?

Had Montresor told of his actions the next morning, the events would have been "a very good joke indeed,"  as Prospero would have been found and rescued; no murder would have taken place.  The last 3 sentences of the story are "Against the new masonry I re-erected to old rampart of bones.  For the half of a century no mortal has disturbed them.  In pace requiescat!"  These suggest that Prospero was left to die and eventually did, but these phrases connote a deathbed confession, a retelling of the actions took place in the character's youth, and this has been the first telling of those events.  It also suggests that Montresor has visited this spot since the entombment, or has sealed up the entrance to the vaults, otherwise he wouldn't have known that the bones he re-erected in front of his masonry hadn't been disturbed. Finally, the last sentence's Latin translates in the familiar "Rest in peace,"  but it's ambiguous as to whom Montresor is referring -- to himself at at advanced age facing death and seeking forgiveness for his actions many years ago, or to his friend that he murdered, or both!

Why does Montresor wait fifty years to tell his story? Would the story differ if told the morning after?

The main reason that Monstressor waits fifty years to tell his story is because he is extremely proud of the fact that he has gotten away with it for so long.  He stresses this point at the end of the story.  This builds into the evilness of the character because the reader already knows that he wants to get his revenge on Fortunado, who probably did not do anything worth being murdered for.  But, at the end of the story we see that Montressor is extremely pompous and wants everyone to know that he has been able to hide this murder for half a century.  If this story had been told the next morning, there may still be the possibility that the murder would have been found out about and the evilness inside of Montressor would not have been stressed as much. 

Why does Montresor wait fifty years to tell his story? Would the story differ if told the morning after?

Had Montresor told this story the morning after the crime, it would have been treated as a confession, his victim would have been found, and he would have undoubtedly been arrested and either sentenced to death or life imprisonment.

As the story is, because it is told 50 years after the fact, one might say that the "statute of limitations" has expired on Montresor's alleged crime. That is, because it happened so long ago, we can still treat is as a very wrong deed, but its importance as murder is diminished by its age. Further, we are led to question the accuracy of the account because of its age. How are we, the readers, supposed to know if Montresor is remembering everything with exact accuracy? By "aging" this tale, Poe has accomplished a unique effect.

Why does Montresor retell his story fifty years after the event?

It is not exactly clear under what precise conditions Montresor is recounting the events of that terrible day. It may be some kind of confession, or it may not even be a true story. Who knows? The important thing is that Montresor did not simply want to wreak revenge on Fortunato; he wanted to get away with it. And with the passage of fifty years since he committed his act of murder, we can be sure that he now feels confident enough to admit what he has done. There is no doubt that, after all these years, Montresor still feels immense pride at getting revenge on Fortunato. But at the same time, he wants to make absolutely certain that he is not going to have to pay for his crimes.