Bottom line is they do pretty much the same. I'm working on a personal project that i have no plans on selling to the market and i only use Windows operating system. So i use DirectX.
If i was to make a game with a game engine i'll make from scratch and try to maximize the reach of my game in the market(different platforms), i would likely implement my rendering engine using OpenGL first.
If i have a lot of money, i would hire a team to develop my rendering engine capable of using DIrectX and OpenGL then use the appropriate grpahics api for the platform my game is running on; DirectX for Windows and OpenGL for other platforms.