Our company is attempting to achieve CMMI ML5, and we would like to know what the difference is between a Capable Process and s Stable Process in High Maturity?
I often get inquiries from organizations that are "going for ML5" and then ask these types of questions. Before I answer it, you may wish to consider that the question itself is an indicator that you might only by "going for ML2 or 3" right now. The CMMI is all about indicators, or feedback concerning your current state, and the question indicates you may not have gone through the prerequisite steps to get to ML5.
While neither term is specifically tied to "high-maturity" it would be difficult to know if you have achieved either a capable OR a stable process without having at least implemented Organizational Process Performance (OPP), as this gives you the information you need to answer the question.
A Capable Process is a process that has achieved it's specified quality and product goals and objectives. A Stable process is one that is operating within specified control limits and without excessive variation. Obviously, statistical process control is required to understand this.
Practically speaking these could be the same thing, or, given the way organizations mature and adapt process over time, they could start out far apart and come closer together over a span of time - often years.