AMD are supporting open standards like OpenCL, which is effectively their answer to CUDA. The problem is that Nvidia refuse to keep their drivers up to date with the latest OpenCL capabilities. The latest version nVidia supports is OpenCL 1.1, but its now up to OpenCL 2.2. nVidia are years behind on OpenCL. This is pretty much the primary reason why CUDA seems so far ahead.

nVidia released new features and capabilities on CUDA that later versions of OpenCL also supported, but were not available on nVidia cards due to lack of OpenCL driver support. Hence, lots of people developed libraries and tools for CUDA, with great encouragement and support from nVidia (which attracted developers). These tools were useful and powerful, encouraging others to also use nVidia hardware to take advantage of those tools. CUDA has now taken root.

If nVidia were to start keeping their drivers updated with the latest version of OpenCL, i suspect CUDA's dominance would start to wane. Though the CUDA roots have gone deep, so it may take several years to reverse its dominance. However, its highly unlikely that nVidia will update their drives any time soon. The only thing that would probably force them is AMD gaining market dominance for a few years.