AFAIK, Tesla just wanted to work for Edison and build new things. They disagreed about a lot and their personal styles were at such odds they were destined to never get along. But, again AFAIK, what really tore it between the two was (to quote an article I found elsewhere):
Tesla insisted that he could increase the efficiency of Edison’s prototypical dynamos... Edison ... promised him $50,000 if he succeeded. Tesla worked around the clock for several months and made a great deal of progress. When he demanded his reward, Edison claimed the offer was a joke, saying, “When you become a full-fledged American, you will appreciate an American joke.” Edison offered a $10/week raise, instead. ... Tesla quit ...