These two positions are relatively similar to one another, which is one reason that many people who had been governors (Jimmy Carter, Ronald Reagan, Bill Clinton, and George W. Bush) have become president in recent times. In both cases, they are the heads of the executive branches of their respective political entities. Therefore, they both have to do things like making budgets and negotiating with legislatures to try to get their agendas passed.
The differences arise mainly from the fact that the president is the head of the larger executive branch and one that has more powers. One example of this is that the president has the power to appoint justices to the Supreme Court. State supreme courts are more typically made up of elected judges, thus depriving the governor of a power the president has. More importantly, the president has to conduct a foreign policy while governors largely do not. Governors do not have to worry about things like how long to leave how many troops in Afghanistan. They do not have to formulate a policy towards the rebellion in Syria. This lack of foreign policy powers and responsibilities is the major difference between the two types of chief executive.