What a question! LOL!
What I say might bring other opinions and dance is dance and you take it to the highest level of artistry possible regardless of style. That said (your taking foxtrot/rumba/swing):
1. I dance International and have that bias although I've taken lessons with the Canadian Professional Smooth Champions and think they're incredible.
2. My limited understanding of American Style was that it was popularized through AM studios and similar as a way to teach people to dance quickly. Most people found International more difficult. Again, that said: Over time American Style developed to a high level of artistry beyond it's original intentions.
3. In the standard dances, such as foxtrot, the most obvious difference is the hold. In International it is very rare for partners to let go of each other except possibly at the beginning or end of a routine. In American my perception is that partners will go to one-hand and no-hand holds and often pose side by side, etc. (Apologies to the American style dancers if I'm getting this wrong). I do not like watching American style on TV however, live, I think it's terrific.
4. American rumba seems to have started as a multi-purpose dance. In International the movement of the feet begins on the 2nd beat of the bar. For beginners this can be very difficult as most people find the strong (1st) beat of a bar more easily than an off-beat. In International you learn to "settle" the hip on the one and move on the two.
I teach beginners to "hip, 2,3,4, hip, 2,3,4..." All even beats.
As many beginners find this difficult American rumba (apologies and again to my limited experience of American style) seems to have been developed out of a simple box step with movement on the 1 (instead of the 2) with a basic count of: 1, 2, and , 3, 4, and... repeated (or Slow, quick, quick, slow, quick, quick). I often see people dancing American rumba in social settings to music that is not strictly rumba. Slow music in general often seems to work.
Regardless of style treat what you do as high art.
Good Luck!
OB (the Know-It-All)
