There is a very big and important difference.
That difference leads to dealing with a number of issues and questions that are plaguing our modern society in addressing many serious new problems today
Basically, the difference is quite simple.
Government is only ONE arm of modern society and it derives its legitimacy and powers from its taxes, spending, laws, and regulations.
The other two arms are:
 the business/for profit sector, which derives its power from creating jobs and paying taxes
 the nonprofit sector, which derives its support by serving the public interest without profit.
Those three, legs or sectors, were never explicitly invented, hence why their exact distinctions are sometimes blurry and/or porous. Note that we are only talking about the three basic sectors as distinguished, for example, from specific sectors like the auto and telecommunications sectors.
To reiterate, the difference between government and governance is that government is only one part of the governance of managing our large and complex modern society. While it is important to clearly understand the distinct roles of the three sectors, it is also very important not to conflate them to avoid distorting their respective roles.
The three sectors each have their own languages, practices and cultures. They often train their ‘members’ to distrust the other sectors, to be able to threaten or limit one another. The government sector can regulate and even command the others, or cast them as adversaries to prevent alleged abuses. The business sector generates economic growth and, when frustrated, can turn loose public opinion and their lobbyists. Finally, the nonprofit sector warns that the others must heed their admonitions or be left without their help in crises.
When folks wear a jersey that says ‘government’, they often think that they serve the whole public interest. In fact, often they are ONLY a tiny piece of the larger public interest. For example, the role of the SEC obviously is minor compared to the larger whole of the public interest.
Simply put, governance is the overall process of integrating and managing a modern society.
OK. Now how can we provide better governance?
One of the biggest problems we face today is that fewer and fewer problems can be dealt with in one sector only.
A nice example of a single sector solution occurred a few years ago in DC when the Motor Vehicle Bureau was revamped and citizens were saved endless hours waiting in line to renew their driver’s licenses, etc. But such single sector solutions are few and far between.
Multi-sector problems are frequently quite complex and very hard to untangle. Peek at http://www.intersector.com and you will see case examples and tools used to create solutions, from combating air pollution to improving public schools.
The bed rock issue is a notional ‘space’ [the intersector] between the three basic sectors where collaboration among the sectors is ESSENTIAL though often difficult to achieve.
Imagine the molecules of the three different sectors in three different colors, and then go further and imagine the intersector space filled with molecules of all three colors competing. Failing to help those molecules get organized and collaborate can lead to them metastasizing into serious cancers in our society.
This is exacerbated by the sectors’ reluctance to trust one another. They must each understand all the time that they are only a part of the governance we need.
Often, ‘nature/leadership’ may help those molecules to organize themselves ‘on the same page’ and help folks work out their differences.
We face today at all levels of government and governance the challenge of getting all these parts to work more smoothly together.
The essential starting point is for more of us to understand the differences between government and governance AND to recognize that the intersector is a very important place to look for some systemic changes that may help manage the governance of our modern society.