Define the difference between Core and Abdominal training?

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.

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.

In my opinion, Core training includes working your abdominals along with working you other core muscles.Your core muscles supports your spine and helps with things like balance. Core muscles are connected to your legs, effecting the way you stand, squat, sit. It’s not just about the abdominal muscles, but also training your back muscles, glutes, and the entire area that connects to your spinal cord and helps your body support your spine.
I think of abdominal training as just focusing on the front side of your body, ignoring the other aspects of core training.

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!

Everyone pretty much knows the answers regarding ABS Training. Core training is a whole different ballgame. CORE is defined as the lumbo-pelvic-hip complex, the thoracic, and cervical spine. There are over 25 muscles that made up the core. An even better question, I think Kurt would be "What is the function of the core?" Once you know the answer to that, then you can see the vase difference between the two. The core's function is to provide dynamic stability during functional activities

Just remember that ab training is all that is. Core training, as you know, involves the abdominal muscles and those of the lower back.

Even if a person says they want ab training, they may not be aware or understand the need to train the entire core. Always educate them on that and program for the core, rather than just the abs.