Looking back, DMC2 probably would have been better received story-wise had they made it several games down the line instead of when they did.
In DMC2, Dante is noticeably older, has seen a lot of shit, and his earlier fun loving attitude would've been worn down pretty hard by all the shit he's been through.
The story had other flaws of course, like recycling Trish with a new character that basically was the same in concept, a flat, one dimensional overarching villain, and the story took itself too seriously for its own good as opposed to the first game, but I think what really took a crap on it was how Dante was portrayed.
With the benefit of 3 and 4 on top of 1, Dante's progression from wisecracker to a lot more emo makes some more sense, but at the time of DMC2's release his transformation from his DMC1 portrayal was pretty jarring.
Shin Megami Tensei Nocturne actually got some decent use out of DMC2 Dante, and the setting and circumstances made Dante's far more serious nature from DMC2 work a lot better, and the implications of the backstory set up a better reason for Dante to be much more serious, while stock DMC2 did not provide a decent character development bridge to justify things.