TITLE: Characterizing graph classes by the same forbidden set in two ways
QUESTION [5 upvotes]: Kuratowski and Wagner theorems characterize planar graphs in terms of forbidden homeomorphic subgraphs and forbidden minors, respectively. It turns out that both forbidden sets are the same: $\{K_5,K_{3,3}\}$.
Are there other examples where a forbidden set defines the same class of graphs in both ways?

REPLY [6 votes]: One example that springs to mind: A (simple) graph is a forest iff it has no $K_3$ as a minor, and also iff it has no $K_3$ as a topological minor (homeomorphic subgraph). 
More generally, if the forbidden graphs have maximum degree 3, there's no difference between minors and topological minors (see Proposition 1.7.4 in Diestel's textbook).