Saying "I don't want to live" often reflects feeling exhausted, empty, or hopeless, or feeling unable to bear a current situation or emotional pain. It usually means wanting the hurt to stop rather than wanting life to end. It can express a desire for relief or escape, not death itself. People who say this often wish for their life to change, not to disappear.
Saying "I want to die" typically conveys a more direct belief that death is the only solution, along with a sense that no alternatives or improvements are possible. It can indicate more active thoughts about ending one's life.