Governance is all about IT making good decisions for, and with the business. At the end of the day, your IT function helps your business make money, so it's vital that they communicate effectively, share information, and collaborate to ensure that their goals are aligned, and that the best use of IT resources is being achieved. The best way to do this is to implement an Governance forum, where representatives can meet to discuss the achievement of strategic goals, items for endorsement or discussion, policy, IT issues, or whatever discussion items to facilitate good decision making.
Compliance is about meeting any regulatory requirements for your IT operation. Largely this is seen in Government, but many enterprises are concerned with the performance of their IT investment, and they will implement their own internal compliance program to ensure that they are operating is a manner that is consistent with good or best practice, and meeting federal privacy requirements. It is important to scale compliance programs according to the size of the organisation, but it's also equally important to ensure that the most important functions (e.g Information Security) are not overlooked.