Hi,
The 2 terms are often used interchangeably. I think most people would see them as the same thing.
That said when we talk about "strength" in a scientific way we mean a muscles ability to create force. With stability we mean how the muscles work to maintain a stable spinal position. So if you use that with the core, stability would mean how well you can maintain a plank or balance on gym ball. Core strength would mean,for example, how much weight you could lift if you did a resisted stomach crunch.
The two are undoubtedly interlinked. With stability the focus is on control and prevention of unwanted movement. The aim is to keep a small arch in your lower back and the pelvis in a neutral position. This can be done while the body is static (e.g. In a plank) or while the body is moving (e.g. Bridging) the core is kept stable throughout. This is one of the principles behind Pilates. It can also involve a contraction of "core muscles". These are deep in the abdomen and pelvic floor. They wrap around the lower back like a corset to help control movement. They include a muscle called transversus abdominus which was a major thing in core stability up until recently when research started to question if it had a role in stability. People will often say to "engage your core" it is often this muscle they mean. A common way to engage it is to gently draw your abdomen in about an inch. The aim is then to maintain this while you do your stability exercise. Stability is about control and quality of movement.
Strength is about load. To work core strength then you could load muscles supposedly involved in stabilising the core. These include glute med and max, rectus abdominus, your obliques, erector spinae and lats. There is some debate though as to whether these muscles have a stabilising role as they are not designed for that. They are more for moving loads than holding the spine in a stable position.
Sorry for such a lengthy answer, it's actually quite a complex question!
