Poster is probably using the terms how they are used in Analysis. A sphere is just the surface, i.e. the set of points equidistant from some center. A ball is the entire volume enclosed by said sphere.
To expand on @contravariant's answer a bit, let me take a more hand-wavy intuitive approach.
The surface of a ball, i.e. a sphere, doesn't have boundary for the same reason that the surface of the Earth doesn't. You can't travel along it and eventually reach some kind of edge. That is, no matter where you are on the Earth, you can in principle, move freely along any coordinate direction and still stay on the surface of the Earth.
Thinking of the entire volume of the Earth, however, is akin to considering a 3D ball. Inside the earth, magma/rocks/etc can, in principle, move freely along all 3 axes. However, once we're at the surface, one of our axes of movement is restricted. We can't go further "out" and still stay within the ball (of the Earth).
As @contravariant hints at, the above can be made precise by using the mechanics of topology to get something called a (topological) manifold. If you're at all inclined, I highly recommend taking some time to get familiar with topology. It's an amazingly deep field with applications in almost any field you can think of!
Without going into extreme depth A sphere is a 2D manifold since any point has a neighbourhood that looks more or less like 2D space.
A ball is like a 3D manifold for the same reason, but it also has some points where you don't have a neighbourhood that looks like 3D space but rather one that looks like half of 3D space. That region is the boundary, so to be exact a ball isn't a 3D manifold but a 3D manifold with boundary.
