A 2-distance $k$-coloring of a graph $G$ is a proper $k$-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of $G$ is the minimum $k$ such that $G$ has a 2-distance $k$-coloring, denoted by $χ_2(G)$. In this paper, we show that $χ_2(G) \leq 20$ for every planar graph $G$ with maximum degree at most six, which improves a former bound $χ_2(G) \leq 21$.