A topological minor of $\Gamma$ is a graph, obtained form a subgraph of $\Gamma$ by collapsing paths of degree-two vertices to single edges
A minor of $\Gamma$ is a graph, obtained form a subgraph of $\Gamma$ by arbitrary edge contractions.
It is not hard to see, that any topological minor is a minor (however the converse is not true).
From the combination of Wagner theorem and Kuratowski theorem, it follows, that any graph, that contains $K_5$ or $K_{3, 3}$ as a minor, contains $K_5$ or $K_{3, 3}$ as a topological minor.
Does $K_5$ being a minor of a graph imply it being a topological minor of that graph? Or does there exist a graph, that contains $K_5$ as non-topological minor, and $K_{3, 3}$ as topological minor?
The following graph has $K_5$ as a minor, by collapsing the red edge, but not as a topological minor, since it does not have the necessary five vertices of degree at least 4.
Note that this graph has $K_{3,3}$ as a topological minor, obtained by deleting the blue edges.