Abdominal training is just that – exercising the abdominal muscles. Whereas “core” training incorporates the abdominal muscles but includes so much more (e.g. lower back muscles). You can essentially work the abdominal muscles without working the rest of the core (isolation), but a good core workout “includes” work on the abdominal muscles.
I hope that this answers your question.
I would answer that it would be very limiting to only think of the abdominal muscles. ‘Core’ in my definition includes everything that is not an arm, a leg or the head. I also include the connective tissue (fascia) in the consideration in an attempt to balance the various myofascial lines (as in Thomas Myers’ “Anatomy Trains”) thorugh MELT techniques.
As Sara said, it is a good opportunity to educate the client about the benefits of this more holistic approach.
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.
There is no separation between these two. The real issue is that previous science has updated to see that the old school crunch (or worse sit up) is not that effective to train the core. This is because it has been found that the rectus abdominis is not as important to stability as previously thought. But now we know that all of the spinal stabilizers need to be trained and that the rectus abdominis is a minor player in that arena.