Although the abdominal muscles are included in the core, it is not the core. The core is the base of the body. The core provides most of the balance and stability for the body. A weak core is a weak body. The core is made up by the abs, pelvic floor, part of the lower back, glutes, hips, and hamstrings. These six parts working in congruency makes a strong body. If any one of these six parts is weak, or not activating enough, it will cause the other parts to overwork resulting in pain and discomfort.
The next time you go work on the core, remember to add in exercises that implement more than just the abs.