A ball is spherical; it's shaped like a sphere — a three-dimensional version of the two-dimensional circle.
The ball image is a good one since the word sphere comes from very similar-sounding Greek and Latin words, both of which mean "ball." The word sphere, however, has much broader usage: we use it in to discussing astronomy, our earth, even our social rank (as in "one's sphere of influence"). Spherical, by contrast, has a much more limited use. It basically is a description of shape: if it's spherical, it simply looks like a ball.