Your core is more than just your abdominal muscles. Indeed your abdominal makes up the majority of the core; with the rectus abdominus, external and internal obliques and the transverse abdominus. Your lower back and glutes; maximus, medius and minimus also contribute to your core stability. DO not forget that the top of your quads and hamstrings also play a part.
There is a difference between core exercises and ab exercises. The former works these muscles in unison strengthening them as they work together (more functionally driven strength), the latter works the ab muscles itself, which is necessary to be able to maintain form in core exercises. Both are important in overall stability and strength.
And example of a core exercises is a plank, where as an ab exercise is a crunch.