The first and biggest thing that jumped out at me was the difference in timing. Obviously there's a basic difference in that foxtrot is danced to four-count music and waltz to three-count, but that doesn't come close to telling the whole story. The thing that always strikes me when I watch foxtrot is how much room there is for idiosyncratic timing, compared to waltz. Most foxtrot music has swing, and a tick-tock approach would obviously be out of place. But I've noticed that no two couples have exactly the same perception of how to divide the beats... they're all sliding the beats around, maybe hitting the 2 a bit early and the 3 a bit late, to try to express that aspect of the music. And the good couples seem to know how to play with that during the dance; they dance ahead of the beat in certain places and behind the beat in others, to an extent that really isn't practical in waltz. Here's a link to a Youtube video that illustrates a few things; it's from the USA Dance Nationals of 2007.http://www.youtube.com/watch?v=A3GQjVQqenY&feature=related
Note the couple #230 at the start of the video. That poor guy just cannot make himself wait for the down beat. Instead of fighting it, he's decided to go with it. (I personally am not a fan, but he has obviously made it work for him.) The #265 couple (the lady in the white and purple dress) has a more measured approach. It's quite a difference after you watch it a few times. In neither case do you see the couple dancing "four on the floor" like you might see in waltz. One thing I always have to think about in foxtrot is how to work with the timing without making the dance look choppy. In waltz, that's not a problem because the demands of the music make the movement almost automatic.
The other thing I've noticed is that the good foxtrot dancers tend to be a bit more restrained with rise and fall than is typical in waltz. They will dance some steps almost flat, and you won't see a huge rise out of them unless they are going to hold the figure at that beat. Actually, this seems to be a bit more of a thing in American style; the International foxtrotters still seem to be a bit more traditional with rise and fall. But they still don't emphasize it quite as much as they do in waltz.