Let $T$ be a tree. A vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. A graph is said to be $K_{1,4}$-free if it does not contain $K_{1,4}$ as an induced subgraph. In this paper, we study the spanning trees with a bounded number of leaves and branch vertices of $K_{1,4}$-free graphs. Applying the main results, we also give some improvements of previous results on the spanning tree with few branch vertices for the case of $K_{1,4}$-free graphs.