One of the most important aspects of Persona 4 is the tough balancing act between the countless different activities you need to accomplish within the limited time the game gives you. Just as in Persona 3, you need to attend class, hang out with friends and raise Social Links, improve the hero's attributes through a wide variety of activities, and still somehow find time to dive into the dungeons in order to fight the Shadows. However, Persona 4 is far from being a carbon copy of its predecessor, and features a number of changes and improvements to the system. If anything, just figuring out what to do each day might be an even more complex challenge than before.
One of the most important and visible changes to daily life in Persona 4 is the addition of changing weather effects. This plays a pretty important role in the plot, thanks to the Midnight Channel (which only appears on rainy nights) and the looming threat of the fog (which marks the deadline for reaching the end of every dungeon), and it plays an equally large role in the gameplay. On rainy days various people change their schedule, soccer practice is cancelled, restaurants change their selection, the main store offer discounts, the effect of studying at the library improves, different fish show up at the fishing spot, rare monsters appear in the dungeons, and countless other minor things all occur, making a rainy day totally different from a sunny one. This adds a level of complexity on top of the basic weekly schedule that drove Persona 3, adding some variety to an already interesting system. It is a bit of a shame that the distinction is so binary, though. Simply adding some more minor changes to cloudy days, or introducing windy days or something, would have helped. Certainly, foggy days should have been made more visible and notable, considering their importance to the plot, but as it is you hardly even see fog even on the days when it is actually important to the story.
An equally important change to daily life compared to Persona 3 is the shift to the characters venturing into the game's dungeons in the daytime, rather than the evening. Now, you need to give up the most important phase of the day in order to progress through the game's dungeons, rather than a lesser one, and now going into those dungeons even leaves you too exhausted to do almost anything in the evening, as well. You could choose to go into Tartarus pretty freely in Persona 3, since you never lost too much by doing so (there was always more time to do thing in the evening than there were things to do), but you have to give up important opportunities to build up Social Links every time you dive into Persona 4's TV World. This forces hard choices, and encourages the player to go through the dungeons in as few trips as possible. At the same time, many Social Links are reduced in availability or are even completely unavailable unless you have already completed the latest dungeon, so there is also an added impetus to clear through the dungeons quickly, which both balances out the impetus to wait and build up Social Links in order to get the added strength and changes the tome of the game quite significantly from how it was in Persona 3.
Thanks to the condition/fatigue system, Persona 3 made it impossible to clear through the dungeons in a very small number of trips and a short amount of time. You could only go so far through Tartarus before your characters became tired and had to leave, and whenever that happened it could take them several days to recover, especially if someone became sick. Persona 4 changes that, though, and completely abandons the condition/fatigue system. Even if you push your characters to the limit in the TV World dungeons, everyone is guaranteed to be in top fighting form the next day, so the only limit on how often you go into the TV World is the opportunity cost, which has been made more important. This places a lot more burden on the player to decide when is the best time to dive into the game's dungeons, unlike in Persona 3 where 90% of the time your best choice is to simply go into Tartarus every time you are in great condition and have a full team, without hesitation.
One final thing I like that mixes up the schedule is the addition of the Persona Forecast system. This system adds special rules and benefits to the Persona Fusion system depending on the day, and also lets you know what the special rule or benefit will be for the next day. In addition to tying in well to the game's weather theme, this can be a strong impetus to either try to raise a Social Link or dive into the TV World, depending on your situation and the exact benefit, which can either make your choices a bit easier or a bit harder, and certainly makes things more interesting. I do wish that they would let you see the forecast a bit further into the future, though.
Since this post is already pretty long, I think I will need to write about the activities you actually do during the day some other time.