The two are different. Both are important! But one should precede the other. First, let's define.
"The Core" is the region of the body between the shoulder girdle and the knees. This comprises three main regions, the upper core, middle core, and lower core.
The upper core consists of the muscle that stabilize the scapulae and rib cage: serratus anterior, pectoralis, middle and lower trapezius, latissimus dorsi and intercostal muscles. These maintain scapular position, open rib position for free breathing, and essentially hold up the upper body.
The middle core consists of the oblique and transverse abdominals, the rectus abdominis, the QL, diaphragm and psoas. These support the abdominal and pelvic viscera and protect and move the lumbar spine.
The lower core consists of the pelvic floor, gluteals, piriformis, TFL, quads and hamstrings. These muscles move the hip joints, support the SI and pelvis, and the quads and hamstrings move the knees as well.
Core strength refers to the ability of these muscle groups to generate power--the ability to push against resistance as in lifting weight, or generate an impulse as in jumping or running. These are important functions.
Core strengthening exercises include the various squats, deadlifts, lunges, situps and crunches, hamstring curls and leg extensions, bench presses and rows.
Because I am about to say that core stabilization is both different and more important, I have to also say that I know that core strength is important, useful, and worth working for.
But core stabilization is both different and more important.
Core stabilization refers not to the ability to produce power, but the coordination of the muscle groups, the ability for you to sense your body and make decisions—both conscious and unconscious—about how to stand, how to sit, how to move, how to lift, what muscle groups to engage in what order and to what ends.
Core stabilization exercises focus on both sensation: can you feel your hips and lower back and how they move together, and expression: can you maintain a relaxed neutral spine while fully flexing and extending your hips and knees? Can you keep your shoulders properly postured with long collarbones and ball securely seated in socket while you push, pull, rotate, and circumduct?
If you choose to push heavy weight simply to push heavy weight, without sufficient regard to muscle activation, posture, without sufficient proprioception to make choices at all, you are prone to injury.
How can you know if your core is stable?
Supine hook lying, spine neutral, toe taps and psoas lifts.
Lateral hook lying, pelvis neutral, clam shell and hip circles.
Paloff press with scapular stabilization.