I am not entirely sure what type of answer you're looking for but one difference is that on a sphere, the two degrees of freedom are somehow more coupled than in a torus.
On a sphere, you can do a half revolution along one degree of freedom, and another half revolution along the other degree of freedom and end up back where you started.
On a torus, this is not the case. You will end up much "farther" than where you started.
To use the robotics examples, a spherical pendulum can return to its rest position after rotating upwards 180 degrees by rotating down 180 degrees in any direction. With a 2D robot arm, rotating one of the arms will never move the other arm, and so rotating arm 1 360 degrees results in a different state than rotating arm 1 180 degrees and then rotating arm 2 180 degrees.
Also see https://en.wikipedia.org/wiki/Hairy_ball_theorem. Smooth and continuous vector fields exist on the torus but not on the sphere. This has consequences for controllers on those spaces.
They are also different using the standard math definition. A sphere cannot be continuously deformed into a torus.