I already speculated on this:
Vergil is ashamed for some reason... maybe it's because he recently stole the Yamato from Nero? Or maybe he's carrying an infant Nero to drop off at the orphanage and feels remorse.
When he was defeated by Dante in DMC1, his armor fell off. In DMC3 he was badly wounded, but brought back up by his Nero Angelo armor (that Mundus provided).
When his armor came off, he felt the pain the Rebellion left him again and he was left out on the streets, with nothing but a cloak to fend off the cold.
Thus, he's cold, he's in pain, he's in emotional turmoil. All three of these things are demonstrated by a ratty-looking cloak his past self would be too proud to wear, otherwise.