The sphere and the torus are the two surfaces every course in geometry starts from, because between them they cover the two cases that matter: a closed surface with no hole and a closed surface with exactly one. Everything else with a smooth, closed, two-sided skin is one of these two stretched, or a torus with more holes added. The sphere is the shape that encloses the most volume for a given amount of surface, which is why soap bubbles, raindrops and small planets all settle into it; the torus is what a circle sweeps out when it is carried around another circle, which is why it turns up whenever something goes round twice at once.
Curvature is what separates them. Every point of a sphere bends the same way in every direction, so no flat map of the world can be honest — every projection has to stretch, tear or shear something, and cartography has spent centuries choosing which lie to tell. A torus bends both ways at once: on the outer rim it curves like a sphere, along the inner rim it curves like a saddle, and around the top and bottom circles it is momentarily as flat as a sheet of paper. Averaged over the whole surface those two kinds of bending cancel exactly, which is the reason a torus, unlike a sphere, can be given a perfectly flat geometry.
Both shapes earn their keep well outside mathematics. Tokamak fusion reactors confine plasma in a torus because a magnetic field can run round and round it without ever hitting an end; doughnut-shaped surfaces describe the state space of anything with two independent cycles, from a double pendulum to the phase of two coupled oscillators; and video games have been wrapping their worlds edge to edge into a torus since the arcade era. The sphere, for its part, is the working model of every planet, every lens and every acoustic wavefront, and the maps and grids drawn on it are the reason a curved earth can be navigated with flat instruments.