Running involves the repetitive impact on the heels and feet. When you are of a heavier build, you are applying more weight and pressure on your feet, and if you are not careful, you could risk injury.
Shoes that have more cushion help reduce the shock and stress on your body. A differential in the heel-to-toe drop will also have the same benefit on your feet.
Most running shoes have a drop of around 10-12 millimeters. If you get a shoe with anything less is moving, you are leaning towards the minimalist style of shoes.
With a lower drop shoe, runners are naturally striking on their mid-foot instead of on their heels. This will send less shock to stressed joints.