Can someone describe what the difference is in "rhythm" and "timing"? And can you please give some very clear examples? Thanks for your thoughts.

Rhythm is the occurance of accented beats in the music which gives the music its character.

Waltz has three beats to a bar, with the first beat accented: ONE two three, ONE two three.

Timing is the allocation of the beats in one or more bars to the steps in a figure, and can include divided beats. Half beats are timed 1 & 2 etc., whereas a 3/4 1/4 pattern (as in Samba or Jive) is given as 1 a 2.

eg Progressive Chasse to R has four steps, timed: 1 2 & 3. Natural Spin Turn has six steps, timed: 1 2 3, 1 2 3.

This may be too simplistic as an explanation, but I've used it in helping novices understand the difference.
Timing is keeping the count on your steps.
Rhythm is moving your body properly to the steps and the accented beats, as Telemark describes.
If someone watching you dance says, "You've got good rhythm," they usually mean your body moved fluidly and nicely to the music. They are not going to say "You have good timing" when they compliment you.
I've always thought that the American style dances were called Smooth and Rhythm for a reason. The American Rhythm dances have the Latin motion and body movements used to the accented beats. Both styles have timing, but one has smooth movement, the other has rhythmic movement.
If I am way off on that analogy, someone who knows a lot more about dancing than I do would have to explain it better.

Thanks so much, Telemark and dheun. Your remarks certainly clarify the subject. I appreciate the examples and especially your clear responses to my question. That's what I was looking for!