For the most part, the words injured and hurt mean the same. They have very similar meanings, but they are not always used in the same context. If you are injured, then you are hurt. Usually, it means that there are bodily problems that happened, whether it be in a car accident or violence.
However, other things could make someone hurt. Their boyfriend or girlfriend could have broken up with you. Then you could be hurt because your heart hurts for your boyfriend or girlfriend. This does not mean that you are injured. Your heart is not damaged or in pain. Instead, it means that you feel emotionally harmful and are crying. Your body does not need to heal, but your emotional state should.