(Disclaimer: I don't dance international style, though my teachers have gone over a little bit of it with me to satisfy my curiosity. I'm sure someone will jump in and correct this if it's wrong.)
American style is in some ways substantially different from international style. First of all, the dances are different.
American smooth is
- Foxtrot, typically a bit faster tempo than international, and there's a SSQQ timing in addition to the usual SQQ one
- Waltz
- Viennese waltz, slower tempo than in international (more on that below)
- Tango, which doesn't look much like international tango
- Peabody, a faster-than-quickstep foxtrot variant, though you don't see it too often
American rhythm is
- Cha-cha
- Mambo -- if you've seen salsa, you have a pretty good idea of what this one is like
- Rumba, slightly slower tempo and a different basic step than international style, as noted elsewhere in this thread
- Bolero, an extremely stylish slow rumba variant (I think it's the prettiest-looking and most expressive of the American-style dances in either category)
- East coast swing, similar to jive at first glance, but with an emphasis on downward rather than upward movement
Some of the stylistic differences are subtle, some are pretty blatant.
American-style Cuban motion involves transferring weight onto a bent leg and straightening, rather than the other way around as in international style. Other than that and some timing differences, an untrained eye would have a hard time distinguishing between American and international cha-cha.
In American smooth, partners aren't required to maintain body contact, the result being that it has a very different feel than standard. There are lots of open and shadow-position patterns that have no equivalents in interational style. Viennese waltz in American style tends to be danced to slightly slower music to allow for that. But foxtrot is often faster in American style, so go figure.
I think tango has a much more distinctive character in American style than in international style, and for my money it's the dance that differs most between American and international style. Lots of dramatic separations and closes. It seems to me that American-style tango is kind of halfway between international and Argentine tango in terms of its feel.
On the other hand, quickstep is much more interesting than Peabody to my eye, lest anyone think I'm being completely provincial here. 
Since there's a lot of open work, lead/follow in smooth can't depend on body contact to the degree it does in standard, though that's less true at more advanced levels.
As to which is more difficult, well, I think it's about equally hard to be really good at either style. I think American is easier in the beginning, by design, to get newcomers out on the social dance floor quickly. But by the time you're past the basics, it seems to me you're going to be working on the same kinds of things either way.
Hope that's new info to some of you!
BTW, if I remember correctly, at least part of my family originally came over from Surrey.