I am now quite a bit further into Persona 4, so it is about time I actually write about the basic flow of the game. Like its predecessor, Persona 4 is built around choosing how to manage your daily activities as time slowly moves forward. You choose when you dive into the dungeons of the game, but whether you choose to do so or not days will still slowly pass until events occur in the plot. However, rather than just using a direct copy of Persona 3's structure, Persona 4 implements the same idea in a very different manner.
The biggest change in Persona 4 is its much greater focus upon the dungeons you explore. In Persona 3, you go into Tartarus primarily to raise your level and gain the equipment you need for the event bosses, but the major events of the game and the battles against the event bosses are controlled by the date (the rise of the full moon), not how far you go into Tartarus. You don't even need to finish climbing the available areas of Tartarus before the next major event. In Persona 4, however, you must reach the end of every dungeon within the alloted time (before the fog sets in), since otherwise you fail your goal and get a Game Over. Instead of being something you do to prepare for inevitable battles that will happen regardless of what you do, going into the dungeons and defeating the boss in the dungeons' depths is your true goal in the game.
The changes made to the system are a big improvement, in my opinion. I once described Persona 3's Tartarus as a "Plot Desert", but because of Persona 4's changes the TV World dungeons are the stage for many of the most important events in the game. You must reach the end of every dungeon within a certain timeframe, so, unlike in Persona 3, the game designers know both when you are going into the dungeon and what characters are available. It also just feels more rewarding and natural to have the most difficult battle at the deepest level of the dungeon be an important story battle, rather than a meaningless battle against a generic foe. The structure of Persona 3 is ill-suited to putting a large number of important plot sequences in Tartarus, but a few slight changes to that structure reverses that flaw entirely.
Another important and beneficial consequence of the change is that it adds to the feeling that the main characters (and thus the player), are actively achieving something by doing what they do. Persona 3 is a great game, but the nature of the way the main characters fight their conflict is very passive. In that game, all you can do is just wait for the next full moon. Even if you complete your climb of Tartarus early, you must still wait for any plot events to happen. Once you fight and defeat the boss that emerges every full moon, the only thing left to you is to get ready for the next full moon, since the characters' only goal is the broad idea of "get rid of Tartarus and the Dark Hour" and every month's battle is only one more small step toward that goal. In Persona 4, though, your goals are almost always much more immediate, so you will either achieve your goal or fail to do so within a limited time-span after the goal appears. Because of this, there are periods of relative peace and tranquility between crisis periods, in which all of your goals are met and there is no new threat and dungeon coming up on the horizon. At the same time, the crisis periods are more focused upon a limited period of time, and the stakes seem higher, so they have a greater degree of tension. As such, the game fluctuates more greatly between periods of high tension (in which your goal is to reach the bottom of the newest dungeon as quickly as possible) and very low tension (in which you are free to pursue a number of optional objectives throughout the older dungeons at a relaxed pace). This helps the game a lot, I think.
Another important part of this change that I haven't touched on is the difference between the goal-posts used for measuring time in the game. Persona 3 is built around the lunar cycle, so every phase of the game lasts around 28 days, which is extremely predictable with no room for significant deviation. Persona 4 is built around the risk of "the fog coming in", which doesn't have a set and predictable cycle. This means that there is a widely variable amount of time that occur between crisis periods in the game, and once a crisis starts you can only really guess at how long you have before you run out of time. This improves the relaxed pace between crises, adds to the tension of a crises period, and, most importantly of all, gives the game designers a lot more flexibility regarding how the calendar is scheduled, which helps a lot regarding things like holiday schedules, unusual events, and exam weeks.
However, mentioning calendars is reminding me of how useful it was having one in Persona 3. The lack of one in Persona 4 is a bit annoying, even though I know it wouldn't be useful in determining how long I have left until the fog sets in. At least Persona 4 more than makes up for it by letting you roll back time if you can't achieve your goal, which prevents the nightmare scenario of being unable to progress any further, and thus having to start the entire game all over again, that is possible (if unlikely) in Persona 3.