@Austin Well Leibnitz did publish his work way before Newton and I assume that was enough. Newton however claimed he had discovered all this using his word as his main piece of evidence. Historian did prove he was in fact telling the truth, that argument wasn't that effective at the time.
Newton started calculus by focusing on differentiation while Leibnitz used integration.
@Joe I couldn't find what the critics said but I know Newton was really sensitive to any form of criticism. Hooke's criticisms almost stopped the production of the third book of Naturalis Principia. 
@Wendy I personally believe the outcome biased in Newton's favor. Poor Leibnitz died a broken man doing number crunching for the first German king of England. Sources say only only person that attended Leibnitz funeral was his faithful secretary. 
@Jordan Well thinks have changed in the a lot in physics. Heisenberg uncertainty principle kinda disproves everything newton worked so hard for. Higher level Mathematician's work is starting to merge with relative physics so math is turning into something relative and uncertain.