It's much more likely that Newton stole Lebnitz' work than the other way around. But the most likely scenario is that both men invented it independently.

However, it is believed that Newton was not going to publish the calculus until he knew of the competition. Newton would have preferred to instead publish the results. I should say that "it is believed" by myself. I recall learning this from some maths history course and am quite sorry that I don't have any references.