What is the difference between Process Stability and Process Capability?

Process Stability refers to the consistency of the process with respect to important process characteristics such as the average value of a key dimension or the variation in that key dimension. If the process behaves consistently over time, then we say that the process is stable or in control. 

Process Capability is a measure of the ability of the process to meet specifications. It tells us how good the individual parts are. There are several methods to measure process capability including an estimation of the ppm (defective parts per million). Capability indices such as Cp, Cpk, Pp, Ppk are very popular; however, trying to summarize the capability via a single index is often misleading because key information about the process is lost.