What is the difference between core training and abdominal training?

Many times, these terms are used interchangably.  This is because the abdominals are part of your core. However, the abdominals do not represent the entire core.  Core musculature surrounds your waist like a weight belt, and while it has dynamic components, it is mainly used to stabilize the spine and pelvis.

Abdominal training includes the shortening or tightening of the abs in such exercises as crunches, V-ups, reverse crunches and leg raises.  Core training focuses on maintaining even pelvic/spinal posture in such exercises as marching on the ball, supine leg slides, single leg work on the bosu or making sure your pelvis elevates evenly when doing a bridge.  By gently pulling your belly button to your spine or doing such exercises as the kegel, you can engage your core musculature.