The OpenGL version is using parallax mapping, where the pixels of the flat plane are rendered by ray-marching into a depth map:

https://github.com/kosua20/GL_Template/blob/3de4e116cdd24df3...

The cycles version, on the other hand, is actually using the depth map to displace a high-res mesh -- a more general technique that should yield results as good or better than parallax mapping. I'd chalk up any perceived differences to lighting and material parameters.