Let $T$ be a tree. A vertex of degree one is a \emph{leaf} of $T$ and a vertex of degree at least three is a \emph{branch vertex} of $T$. A graph is said to be claw-free if it does not contain $K_{1,3}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of claw-free graphs. Applying the main results, we also give some improvements of previous results on the spanning trees with few branch vertices for the case of claw-free graphs.