Comparison
In general, Direct3D is designed to be a 3D hardware interface. The feature set of Direct3D is derived from the feature set of what hardware provides. OpenGL, on the other hand, is designed to be a 3D rendering system that may be hardware accelerated. These two APIs are fundamentally designed under two separate modes of thought.
Even so, there are functional differences in how the two APIs work. Direct3D expects the application to manage hardware resources; OpenGL makes the implementation do it. This tradeoff for OpenGL decreases difficulty in developing for the API, while at the same time increasing the complexity of creating an implementation (or driver) that performs well. With Direct3D, the developer must manage hardware resources independently; however, the implementation is simpler, and developers have the flexibility to allocate resources in the most efficient way possible for their application.
Until about 2005, another functional difference between the APIs was the way they handled rendering to textures: the Direct3D method (SetRenderTarget()) is convenient, while previous versions of OpenGL required the manipulation of P-buffers (pixel buffers). This was cumbersome and risky: if the programmer's codepath was different from that anticipated by the driver manufacturer, the code would have fallen back to software rendering, causing a substantial performance drop. However, widespread support for the "frame buffer objects" extension, which provided an OpenGL equivalent of the Direct3D method, successfully addressed this shortcoming, and the "render target" feature of OpenGL brought OpenGL up to par with Direct3D in this respect.
Outside of a few minor functional differences which have mostly been addressed over the years, the two APIs provide nearly the same level of functionality.
(Erst mit OpenGL Version 3.2 werden Geometry Shader unterstützt, also erst ab 7. Dezember 2009)
