First Impressions
Note: I've played roughly 4 hours of the single-player game (not coop). This is not a review but my short-and-sweet first impressions . Also: the fun is going to be overused in my description of the game.
Graphics
This game looks a lot better than SR1 and SR2, but graphically speaking I don't think SR3 is going to blow anyone away. Technically Volition has managed to smooth things out. There is now more of a steady frame rate, more post processing effects going on, less screen tearing and the animations are better. Overall, it's just more interesting to watch.
What is also wonderful is the wackiness of the game spills out into the overall look and feel of the game. There is a lot of great and wildly fun things going on with the UI; Each mission complete screen is a dramatic screen cap of the action. Every activity seems to have it's own little animation and graphic style.
Sitting back after playing it for a few hours, I realize there was one thought that kept popping into my head about the graphics: Saints Row The Third looks more like GTA4 (graphically!) than I thought it would. The texturing of the streets now look like they have potholes, nicks and are degrading from ware. I seem to remember SR1/SR2 having streets looking like plastic bags taped together. Whatever that means. The point is, the game looks better but it's still lot Skyrim, or even GTA4 (but close).
Also: The UI is snappier and cleaner in design than its predecessors.
Gameplay
I've been focusing on purchasing property and some side stuff to build up my cash flow first so I haven't had too much time with the SP portion, but what I've seen is a lot more interesting and fun to play. The script is now legitimately funny at times, and other times it can be a bit boring and drabble on.
The main mechanic is the leveling system that unlocks more abilities. This level system is based off of respect, which is gathered by doing just about anything that's considered violent or juvenile.
The side stuff is great! One of my favorites so far is the Insurance Fraud where you run around trying to rack up as many points in damage (to yourself) as possible. This is done by hitting one of the shoulder buttons, which makes you ragdoll, and then watching your character bounce around a free and rack up cash. So. much. fun.
The Mayhem missions consist of you driving a tank around town and blowing stuff up as you build a multiplies racking up cash.
There are a few more that are also great but, whatever, just go get this game, turn your brain off and enjoy.
Oh and for those that might be curious: The cussing is still there but it's not as aggressively gratuitous as the previous ones. Oh and the use of Kanye West's Power is perfect--you'll see.
Questions? Shoot!