In my usage, a soliton and a topological defect are synonyms for certain isolated solutions to differential equations. Please replace soliton with topological defect in my statement above and see if it makes more sense.

The idea is there are families of solutions to partial differential equations that are stable, but cannot not deform continuously into any of the "ordinary" solutions without leaving the set of solutions at some intermediate point of deformation. These outlying solutions are topologically isolated Islands in the solution space.

We can think of them as disconnected families of non-standard solutions - solitons.

We can also think about them as topological defects in the set of solutions. We ordinarily expect the solution set of a differential equation to form a path connected set including the trivial solution. If there is some region of solutions that is topologically cut off from the ordinary solutions, we call its solutions topological defects. But these are nothing more than the solutions that can't be smoothly deformed into the zero solution.