You would say someone is obsessed WITH somebody or something. It is specific.
Obsessive is a general personality trait - a tendency to become obsessed easily.
He is quite obsessive. He can't just enjoy stuff normally. He throws himself into it completely.
At the moment, he's obsessed with Taylor Swift. He's listening to nothing but her.
You can also say 'obsessive about' and this is specific but the meaning is different.
He's obsessive about cleanliness = every must be perfectly clean all the time.