--Warning: Spoiler alert. If you don't wanna know what happens in the opening cutscene of Mission 1, do not read this post.--
Okay, quick step-by-step. As Nero is about to leave the opera house, his demon arm glows blue through the sling. He looks up to see Dante crash through a skylight, land on the altar in front of Sanctus and put a bullet in the old man's head. Credo cries out to His Holiness and sets his knights upon Dante. The son of Sparda effortlessly slays them, only to take a dropkick to the face from Nero, who quickly takes the field against Dante. On Credo's orders, he attempts to stall Dante until reinforcements arrive (for all the good reinforcements will do against this kind of enemy), but Dante escapes just as Credo returns with a few more knights. Credo promtly has Nero track down Dante and try to bring him in.
--End of spoiler--
On a side note, if you've see any trailers that show Dante with a bloody face and haven't played the game, that blood is not his own. It belongs to Sanctus, and got onto Dante through his proximity to His Holiness as he shot the bloke.
Basically, Nero is, as these two have said, after Dante on Credo's orders. Besides, if you belonged to a sect of knights who worshipped Sparda and some geezer in a red coat dropped in and capped your leader, what would you do?
Are you ready for the changes?