My thoughts:
Capcom took (almost) everything great about Devil May Cry 3 and stripped it away, and all they gave us in return was prettier graphics.
Whereas in DMC3 we were in control of a young Dante who seems to be channeling the Ninja Turtles and was completely charismatic and entertaining in every cutscene, in DMC4 we get Nero: a whiny, crying, angsty teenager who would be right at home in a teenage anime like Bleach.
DMC3 was cheesy, but in a good way. The movies were very entertaining and served as an incentive to plow through the game's harsh difficulty. In addition it made you really like Dante. Hell even Vergil was a decent character compared to what we get in DMC4. I already covered Nero's character, but his style is lame too. He seems to be challening Kratos and his brutality, except without the over-the-top gore and manly voice all we're left with is an annoying effeminate schitzo.
Nero's fighting style is okay, for the most part it's the same as Dante, except his Devil Arm style which I personally didn't like. It made the game feel more like God of War than DMC. Thankfully we eventually get back in control of Dante and his 4 styles, but he's practically neutered. In DMC3 you got 5 devil arms and 5 guns, here Dante only has 3. On top of that, they took out the "crazy" combos from DMC3 (why??! why would you do that?!) and even gimped Trickster a little bit.
DMC3 had loads more enemy variety and bosses, I think this goes without saying. Also, after all the cool bonuses from DMC3 (the costumes, the Vergil replay) I was dissapointed with the replay value of DMC4. Bloody palace is back, but now it has a time limit for some reason.
In short, DMC4 is a huge step down and I'm wary that Capcom will focus on Nero in DMC5 as well. I hope it doesn't happen, at most he should be a bonus replay like Vergil was in DMC3 (although who wouldn't rather have Vergil instead of Nero? seriously...)
edit: Oh yeah, and DMC4 feels like a kid's game compared to DMC3, way too easy. Although I guess we can blame the current generation of consoles for that. It seems to be the trend to make games little more than interactive movies, requiring only the most basic reflexes and timing.