There were two real weaknesses about the second game, Justice for All, that hurt it compared to the other games in the series:
1) The cases were not solidly linked together either into the meta-plot of all three games (with the sole exception of the second case) or to each other. Characters and plot threads did not carry over between cases as much as in the first game.
2) The dramatic timing and pacing of the cases were much weaker. The drama of Maya's sudden reappearance into Phoenix's life in the second case was weakened by her brief appearance in the first "tutorial" case of the game. An even greater problem is that the real killers are exposed way too early in many of the cases. For example, the real killer in case 3 is explicitly named by the time the second day of trial even begins. At that point, the player is just going through the motions, without the dramatic pay-off. This was a major blow to the fun factor of the game. Most importantly, there were far fewer dramatic revelations blowing the minds of both the characters and the player.
Trials and Tribulations did a lot to correct these mistakes. The most obvious improvement over the last game is in the amount of cohesion between the cases, through the creation of a meta-plot within the game that starts with the first case and builds towards the final case. There are even points where a fine attention to detail in earlier cases can reveal hints to the mysteries in the final case. The addition of a new recurring criminal also creates the best villain in the series since Manfred von Karma in the first game. The cases have also become more convoluted, with a lot more surprise twists and turns. For one example, there is a point in the second case where a witness makes a revelation that catches pretty much everyone off guard. In the final case, I wasn't able to guess what really happened at all until the last part of the last day of trial, and I am pretty good at putting the pieces together. It was a lot of fun.
The creators of the Phoenix Wright series must have realized that the greatest strength of the series lies in its colorful and eccentric cast of characters. Phoenix Wright himself is actually fleshed out much better in the third game than ever before. However, the best part of the game is being able to control other important characters of the series and getting a chance to see the game's world through their eyes and internal monologues. The humor in the game is also easily up to par for the series. They even managed to squeeze in a "This is madness!/ No. This is SPARTA!" joke, as well as a host of other movie references.
Right now, the real trick for the series will be making the transition to a new main character, Apollo Justice, successfully. It is certainly the right time, since Trials and Tribulations wrapped up all of the dangling plot threads in the series so far. While the creators could have kept Phoenix Wright going as the main character, that would require leaving parts of his story unresolved. It is very satisfying to see resolution. Depriving the fans of resolution forever only leads to dragging a game series through the dirt. Resolving everything with the third installment is a good choice.
Fortunately for the game designers, they have a few advantages going into Apollo Justice. For one, they already introduced Emma Skye, one of the new characters, way back in the original DS port of Phoenix Wright, and made her interesting and endearing. Furthermore, it won't be hard for them to give some familiar faces some cameos. They already have been doing so for most of the series. Finally, all it will take will be a few appearances of the Steel Samurai, the Blue Badger, or Mask☆DeMasque to make veteran players feel at home. At that point, all they have to do is make the new main characters sufficiently fun and memorable.