I believe it wasn't as good, because:
a) It felt like you had to repeat half the game all over again just with a different character. Same areas, same bosses. That felt lazy on the part of the developers, even if it does make sense within the story to an extent.
b) If you wanted to play as Dante, you only got half a game as him. If you didn't, and preferred Nero, same there. The action also felt rather slow to me (after DMC3), especially as Nero.
c) The whole "rescue the damsel" thing has never really been Devil May Cry, it has always played a minor role... Devil May Cry is mainly about punishing those devils. While I don't hate on the Nero-Kyrie thing - because in some ways it's refreshing to see a character who is not 'strong and defiant' and a character that is honestly in love or cares genuinely for another in a more tangible way - I don't think it fits too well in the series. It feels like a guilty cliche in that way. Just to mention, I have no problems with Nero's character - just because he cares about Kyrie and cries for her is not that different than Dante crying about Trish in DMC1 or for Vergil at the end of DMC3. Devil May Cry is all about emotional response. All the 'emo' comments and hate about him seemed to be as vapid as the knee-jerk hate for this new Dante when we very first saw him. Nero is a strong character, not a weak one, emotionally or physically.
d) Aside from Nero, the newbie, the character of Dante I felt was overdone. Don't get me wrong, Dante is always "over the top" and a bit silly, except in DMC2, and in that he felt terribly devoid because of it. But there were a lot of theatrics here. Some people preferred this mature (but still daft) Dante... no problem with that. I just didn't go for it this time.
e) Female characters. Let's face it, they ruined Lady, turning her into almost the opposite of what she used to be, gave her a boob job and a shirt without too many buttons this time. I liked Lady's character very much in 3 - my favourite of the DMC heroines because she's very believable. In 4 though... eurgh. Trish and her 'Gloria' persona - while I know Trish is a demon created originally to lure and be alluring - I think I'll just echo Vergil's UMvC3 sentiments on how it's just a bit repulsive. And Kyrie is... well, quite helpless and doesn't get much to do. They all feel rather shallow in this game, and that's a shame because usually the females have a strong character arc and do get something to do. Here, they don't.
f) Loose ends. Who is Nero? Why is he such a mystery? Why are we dangled hints that he's a Sparda relation or a Vergil relation but it's never confirmed? In Western stories, people tend to tie up loose ends and explain things and Western audiences like that - and while I appreciate that in Japanese stories there's a lot more unexplained and simply implied and left at that (and I like that too) the mystery of Nero was done in a way that only ended up bugging the crap out of me. Of course that leaves things open for explanation if they ever made another game featuring Nero... it just wasn't 'tied' up in a way that things were in most of the other DMC games, and it almost feels (when you take the novel into account) that CAPCOM couldn't decide whether to confirm what he was or not, and I'm not sure why that is, beyond that they intended to rile or pique the curiosity of fans.