Dot product serves the purpose of finding what component of a vector is the direction of some other vector. Or if we know the vectors them it can serve the purpose of finding the angle between them. It can be used to test whether or not two vectors are orthogonal.
While the cross product is used when we want to find a new vector perpendicular to two known vectors. Or when we want to find the area of a triangle or a parallelogram if we know two adjacent sides in vector form.