Dante is not immortal. He clearly ages, and he does tire out. We've seen this. Infact, when getting sliced and diced by a powerful demonic weapon(namely the swords of Sparda), he wears out badly. He has limits to his power output.
Alucard is a centuries old vampire, the Unlife King, and the most powerful friggin' thing on the night side of Kal-El. He can walk in day like, it just tends to irritate him since it's not his nature time. He can regenerate from a blood smear, and he NEVER tires. I'm sorry, but stats and powers alone, Alucard would trounce Dante in a prolonged all out fight. And I'm a Dante fanboy saying this. Though as Vi pointed out...they'd have no reason to fight...unless Alucard for some reason went total rogue on humanity. And even then...as Alucard himself said, "It takes a man to kill a monster".
And a vampire trivia note: Sunlight didn't hurt vampires until the 1922 silent film "Nosferatu", where the sun was the instrument of Count Orlok's demise. Even Dracula, in Stoker's book, could walk in broad daylight, but he lacked many of his greater powers since daytime was not his natural time.