This does not pertain to the strength of the muscles or force of their contraction. The muscles involved are also different. Core strength deals with the superficial muscles of the core; the abdominals, the gluteals, adductors, abductors, the spinal erectors and other trunk and hip muscles.ore stability is the coordinated effort of the deep muscles of the trunk, pelvis, hips, abdominal muscles and small muscles along the spinal column.