The obvious difference between branches and roots is their function and location: branches grow from the higher stem of the plant and serve to spread leaf growth, allowing greater sunlight collection; roots grow from the lower stem of the plant and serve to spread underground surface area, allowing greater water collection.
Their functions are similar but their modes are different. Shoot branches are soft infant branches, allowing the formation of new leaf growth. Shoot branches grow from the non-specialized tissue in the Apical Meristem, becoming specialized in their function and allowing the plant to grow taller.
Lateral roots are smaller roots that grow "laterally," or sideways, from the main root, allowing the plant to gather more water. These roots are typically very small and can spread to great distances underground, creating, in some plants, the bulk of plant's total surface area. By allowing more water collection, the plant can ensure that the upper stem and leaves are properly hydrated, allowing greater growth in all areas.