As far as differences, Abdominal Training is typically just engaging the muscles in front of the abdominal wall (esp. rectus abdominis) while Core or Pillar Training uses all the stabilizing and dynamic muscles of the torso (front & back from armpits to hips). Most clients know they want “awesome abs”, they just don’t know the “core” lingo. I usually take this opportunity to explain the benefits of a balanced training program using all the muscles in the core (fewer injuries, more spinal stability, more power for other lifts, better transfer of forces in sports, etc.). Some clients don’t really want to know why, they just want to see results! If they say they want a stronger core, just tell them they’re in the right place and prove it in the workout.