Given a graph G, the {\em maximum internal spanning tree problem} (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of cost-efficient communication networks and water supply networks and hence has been extensively studied in the literature. MIST is NP-hard and hence a number of polynomial-time approximation algorithms have been designed for MIST in the literature. The previously best polynomial-time approximation algorithm for MIST achieves a ratio of 3/4. In this paper, we first design a simpler algorithm that achieves the same ratio and the same time complexity as the previous best. We then refine the algorithm into a new approximation algorithm that achieves a better ratio (namely, 13/17) with the same time complexity. Our new algorithm explores much deeper structure of the problem than the previous best. The discovered structure may be used to design even better approximation or parameterized algorithms for the problem in the future.