I will break the down the difference before abdominal and core training to it’s simplest form.
*Abdominal training is executing specific exercises to tone and strengthen the 4 main abdominal muscles. For an example doing a crunch or sit-up. The goal is to develop the famous 6 PACK; ie working the rectus abdominis (front area), external and internal obliques (waist-line areas) and the transverse abdominis (lower area).
*Core training is when not only the ab muscles listed above are targeted but also the erector spinae (lower back muscles), pelvis (hip area), glutues (butt area) at the same time. The goal is to develop better balance and power when executing any exercise/movement using the arms and/or legs. A popular exercise to build a stronger core is holding the plank position and effectively contracting all these muscles at the same time.
In both trainings, controlled breathing and applying proper form during exercises will absolutely produce better results. I believe it’s very important to focus on all these areas so one can have not only the look but the strenght of a sexy mid-section as well!