In short, core training consists of how well you can stabilize your midsection and then generate force from it. Abdominal training is limited to crunching motions which cause your abdomen to contract (shorten).
Here are 7️⃣ great CORE STRENGTHENING exercises you can do with weights, or substitute with household items, like a water bottle, pillow, or backpack:
1. Bent Knee Deadbug
2. Bird/Dog Reverse Fly
3. Plank Walk Outs
4. Straight Legged Deadbug
5. Reverse Leg Raise w Hip Abduction
6. Plank w/ Pull Through
7. Weighted Leg Lowers
Most of these exercises focus on your ability to resist moving your spine while moving your limbs (core strength). It is important to start slow with minimal weight and reps and work your way up from there when the proper form can be maintained.
For the leg lowering/raising exercises make sure to only lower the legs as far as you can keep your low back pushed to the floor. Keeping your back floored is the key to all of those exercises. Take it slow and modify your range of motion when you first start. Only go as deep as your body allows.