A midwife is primarily concerned with assisting a mother during the birth event itself and is not generally involved much before or beyond that time. A Doula however, can be involved for as much or as little of the process as you desire (see Types of Doulas for more details).
So, for example, maybe you didn’t have a Doula for the birth but during the postpartum period you find yourself needing more assistance and hand’s on care. A good postpartum Doula can help you with any of the issues/challenges you are experiencing.