Behaviour is the way someone acts in a certain situation, e.g. "His behaviour that night was disgusting". Maybe the person's behaviour is different when they drink alcohol or they are around different people.
A trait is something associated with a person, perhaps a person has a lisp, or a limp, or they laugh in a strange way. That is trait that belongs to that person's personality. This is also called a personality trait. If you are describing a person to someone else, you may refer to their personality traits to help show them a picture of the person.
E.g. "You know that girl you saw last week, with the blonde hair and high-pitched laugh? I'm seeing her tonight".