The quick answer is this: International style features a predominantly straight leg action, while American style features a bent leg action.
It's important to be aware, however, that "straight leg action" does not mean the legs never bend, any more than "bent leg action" means they're always bent. The truth is, both styles of movement use alternating bent and straight legs; What gives the impression of a "straight action" or a "bent action" is simply the timing of the straightening. To produce a straight leg action, the straightening action is completed before the transfer of weight to a foot; To achieve a bent leg action, the leg remains bent through the weight transfer.
Instead of using the terms International and American, I prefer to describe the actions as "pressed" or "extended", since both Americans and International dancers use both types of leg action at various times. With this in mind, I'll describe the action of a forward walk to illustrate the differences in timing between the pressed and extended versions.
Begin standing upright with both legs straight, pointing one toe backwards. With either action, a walk begins with a relaxing or "settling" of the hip towards the standing leg. This is accompanied by a bending of the knee of the free leg. The free foot then moves forward towards its new position, and this is where the pressed and extended walk part company.
For an extended walk, as the free foot begins to point forward having passed the standing foot, the knee begins to straighten. The leg is fully extended to a straight forward point before the transfer of weight begins. For a pressed walk, the knee remains bent as the foot moves forward of the body. Depending on who you talk to, the leg will then either remain fully bent throughout the weight transfer, only to straighten once the weight change is complete, or the leg will straighten concurrently with the transfer of weight, both finishing at the same time.
As you can see from the example, the difference between a pressed and extended action comes down to one specific moment. The remainder is identical. So if you think about it, there is actually more similarity than difference between pressed and extended movements.
Regards,
Jonathan Atkinson
www.ballroomdancers.com