TL;DR: A very bloody anarchy, followed by an intellectual bloom and finally listlessness.

If every day is the same day, law enforcement becomes moot. Sure, you could arrest Fred for robbing the bank, but what's the point? You can only hold him until midnight before he's back on the street. Even if you could rush him through the judicial system, there's no point: he won't stay in confinement. You could repeat the process every day, but how many people are going to be doing the same thing? Because Fred didn't rob the bank, now George can. So arrest George too. Because Fred and George didn't rob the bank, Alex did. So arrest Alex too. Eventually, the police spend all their time trying to keep up with arrests when the criminals are just going to be free at midnight. So, law enforcement goes out the window (don't worry it'll be back at midnight for a brief showing.)

The arts will take a hit as well. Anything that has a material result, such as a painting or a hand-crafted chair, is going to be a no go. Authors, painters, and architects all lose their work every day. Gamers will suffer as well. Know that boss you struggled with for a week that you finally beat? Sorry, but you have to face him again tomorrow. And the day after that, and the day after that... In short, anything that has a material product is out the window (don't worry, this will make a brief appearance each morning, too).

So now we come to the point about anarchy. We have removed any useful application of law enforcement and there's nothing people can do to keep their hands busy without driving themselves insane. So people go out on the streets and (gasp) interact with each other. At least we remember the conversations we had... Oh wait. Jane remembers that Joshua beat her up yesterday, so she's going to take out revenge before he can do it again. This leads to the largest mass killing the world has ever seen. And each day, it just gets bigger. At some point, escalation results in the vast majority of human society vanishing from the face of the Earth each day, at least until it's clear that it doesn't matter how many times it happens and we give up on the concept of "An eye for an eye." When will that happen? Not sure. It's kinda been around awhile, even modern societies have the death penalty (you killed Clemont, so we kill you).

Scientific communities, on the other hand, will be both rewarded and annoyed. The people working in Cern on the Large Hadron Collider are never going to get any more work done, it just takes too long to prepare everything and analyze the massive amount of data that comes out of it. Philosophers will have a field day, however, they can debate to their hearts' content and remember everything that was said from the previous day.

After all the anarchist bloodshed, there will be a growth in philosophy and theology. Death becomes meaningless (except to those unfortunate enough to be slated to die every day) and the value of life vanishes. If this effect happens across the universe, we'll never be able to answer questions like "Is there life out there?" We can develop all the theories we want, but we'll have to start from scratch on everything we produce.

We have no production any more, which leads to an interesting consequence. If there's no reason to make anything, then why go to work? If the majority of the population isn't going to work, then why should the people who run the public utilities go to work? Why should they burn away their day when everyone else is sitting at home watching reruns of Law & Order (not that they aren't already)? It's not like not going to work is going to impact anything. If something breaks, it'll be repaired tomorrow. No problem. So we lose some power generation.

Let's look at the extreme case of power loss, though. Active nuclear facilities are always in danger of going critical. Let's suppose a critical part fails one day. Now we have a new catastrophe and, the next day, the workers are back on the job, much to their displeasure.

After all is said and done - quite literally in this case - all that's left is to do nothing. So, after all the bloodshed, tears, and philosophy, we come to the point where everyone just lazes about, waiting for the next day when they will do the same thing. This'll lead to listlessness and, for those of us with extremely active imaginations who now have no reason to apply that ingenuity, depression. For the more depressed, suicide becomes a daily thing to avoid the daily lack of activity (don't worry too much, they'll be back tomorrow).

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