Perhaps the most obvious difference is that DirectX, as opposed to OpenGL, is more than just a graphics API. DirectX contains tools to deal with such components of a game as sound, music, input, networking, and multimedia. On the other hand, OpenGL is strictly a graphics API. So what aspect of OpenGL sets it apart from the DirectX graphics component?
Well, first things first: both APIs rely on the use of the traditional graphics pipeline. This is the same pipeline that has been used in computer games since the early days of computer graphics. Although it has been modified in slight ways to adapt with advancements in hardware, the basic idea remains intact.
Both OpenGL and DirectX describe vertices as a set of data consisting of coordinates in space that define the vertex location and any other vertex related data. Graphics primitives, such as points, lines, and triangles, are defined as an ordered set of vertices. There is a difference in how each API handles how vertices are combined to form primitives.
There are a bunch of differences in the DirectX and OpenGL APIs, so I will list a few of those for you. This chart is based off the book, OpenGL Game Programming and a few of these may now be incorrect as new DirectX versions are released. If you wish to correct me, please do via one of the ways listed at the end of this tutorial