I am training with DONA and their position is that doulas don't do anything medical. They support the mom as a birth partner. They are trained to help get through the (sometimes) long and intense labor. A husband/life partner are not trained this way (usually).
Doulas don't take blood pressure, heartrates, cervical exams, etc. They are not trained to deliver babies, only to help the mom to be as comfortable as possible while laboring and birthing. Which is a very big job for a husband/life-partner to do alone, and can be more than a midwife can do while she's focusing on the baby being birthed.
A doula is there from the very beginning, through the birth, and into early post-partum. The midwife might just be there for the end of labor, on.
Some midwives are able to be there for a longer period of time and usually do provide emotional support, etc. But usually, I think they come in later on.
This is why doulas and midwives (or OBs) make good teams. They do compliment each other providing lots of love and support.