When walking on a flat plane, your knees don't bend as much. Your hamstrings will do most of the work.
By walking on an incline, your knees bend more and as a result, your quads can be used to extend the knee back to the straight position. The greater stretch on your glutes allows them to generate more power to thrust the leg backward. The more acute angle of your foot will help exercise ranges of the calves which you aren't used to in daily walking.
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You will burn more calories on the incline than the flat plane, granted that both are set on the same speed. This is because you need to generate an additional [mass * gravity * sin(angle)] newtons to move upward. On a flat plane, the angle is 0; sin(0) = 0, so no force is used to move upward.
In conclusion, the incline will work a larger range of muscles. But I don't think the incline is necessarily better than the flat plane. The point of the threadmill is to do cardio, not build muscle. But if the angle is particularly steep, it can be used to increase muscular endurance of those particular listed muscles somewhat.