This article is from the Recreational Figure Skating FAQ, by Karen Bryden with numerous contributions by others.
The difference between these strokes lies in where the new skating
foot sits on the ice at the beginning of the power stroke. In a normal
stroke, the new skating foot is placed alongside the skating foot and
the push is outward and slightly to the rear.
For the cross-over (aka cross stroke, cross pull), the new skating
foot is passed across the front of or over the skating foot and placed
inside and slightly ahead of the skating foot. The push has a strong
sideways component, as if you were "climbing stairs sideways".
In the progressive stroke the new skating foot is placed on the ice
along side the skating foot and then slides to a position forward of
the skating foot prior to the power stroke. While push is still
primarily outward, it has a more profound front to back component.
Note that new skating foot is *not* simply placed on the ice ahead of
the skating foot, which produces an uneven "walking" motion.
A progressive run (sometimes just called a run) is a merely sequence
of progressive strokes along the same lobe. The difficulty is in
making them clean power strokes in time with the music, and
maintaining the edge and aim. Dance students who haven't mastered the
progressive stroke tend to interpret runs as a sequence of short
choppy strokes or a sort of shuffle sequence.
Both the cross-over and progressive strokes can be executed in either
the forward or backward direction. They are more powerful than the
standard stroke because of the extended length of the power stroke and
degree to which that stroke can work against the weight of the body.
Backwards crossovers are extremely powerful and are often seen in
freestyle to regain momentum between moves, while progressive runs are
used in dance to add power without disrupting the flow of the edges.