What is the difference between OpenCL and OpenGL's compute shader?

I know OpenCL gives control of the GPU's memory architecture and thus allows better optimization, but, leaving this aside, can we use Compute Shaders for vector operations (addition, multiplication, inversion, etc.)?

In contrast to the other OpenGL shader types, compute shaders are not directly related to computer graphics and provide a much more direct abstraction of the underlying hardware, similar to CUDA and OpenCL. It provides customizable work group size, shared memory, intra-group synchronization and all those things known and loved from CUDA and OpenCL.

The main differences are basically:

• It uses GLSL instead of OpenCL C. While there isn't such a huge difference bewteen those programming languages, you can however use all the graphics-related GLSL functions not available to OpenCL, like advanced texture types (e.g. cube map arrays), advanced filtering (e.g. mipmapping, well Ok, you will probably need to compute the mip-level yourself), and little convenience things like 4x4 matrices or geometric functions.
• It is an OpenGL shader program like any other GLSL shader. This means accessing OpenGL data (like buffers, textures, images) is just trivial, while interfacing between OpenGL and OpenCL/CUDA can get tedious, with possible manual synchronization effort from your side. In the same way integrating it into an existing OpenGL workflow is also trivial, while setting up OpenCL is a book on its own, not to speak of its integration into an existing graphics pipeline.

So what this comes down to is, that compute shaders are really intended for use within existing OpenGL applications, though exhibiting the usual (OpenCL/CUDA-like) compute-approach to GPU programming, in contrast to the graphics-approach of the other shader stages, which didn't have the compute-flexibility of OpenCL/CUDA (while offering other advantages, of course). So doing compute tasks is more flexible, direct and easy than either squeezing them into other shader stages not intended for general computing or introducing an additional computing framework you have to synchronize with.

Compute shaders should be able to do nearly anything achievable with OpenCL with the same flexibility and control over hardware resources and with the same programming approach. So if you have a good GPU-suitable algorithm (that would work well with CUDA or OpenCL) for the task you want to do, then yes, you can also do it with compute shaders, too. But it wouldn't make that much sense to use OpenGL (which still is and will probably always be a framework for real-time computer graphics in the first place) only because of compute shaders. For this you can just use OpenCL or CUDA. The real strength of compute shaders comes into play when mixing graphics and compute capabilities.

Look here for another perspective. Summarizing:

Yes, OpenCL already existed, but it targets heavyweight applications (think CFD, FEM, etc), and it is much more universal than OpenGL (think beyond GPUs... Intel's Xeon Phi architecture supports >50 x86 cores).

Also, sharing buffers between OpenGL/CUDA and OpenCL is not fun.