As I see it, the difference is in the number of people who are listening to you. In a private situation where one person is shy or quiet and others cannot hear him or hear him well, he might be asked to "speak up". Often it is used to refer to not being afraid to express your opinion.
Speak out has a more sociological reference: A person will "speak out" in favor of one project, or "speak out" against another.
Let's see a few situations where the verb "speak out" is used on the world wide web.
We begin with the title of a book:
Can you explain why the verb "speak out" and not just the word "speak" is used here? What social forces would be upon her to keep her from expressing her viewpoints?
Here's another headline:
The website continues: Medal of Honor Recipients Speak Out About PTS. Why are they using the verb "speak out" and not just "speak up"? Who does the organization want to listen to what the Medal of Honor recipients are saying? What changes does the organization want to be made? Why?
It doesn't have to be a national or international issue for somebody to speak out. Here is a local issue:
Have you ever spoken out on an important social issue? If so, how? To whom? Are there any issues that you would like to speak out about, but haven't found the time, the proper means, or "your voice"?