Boiling up means starting to boil. I guess it could also mean that it is boiling, but not yet finished cooking. Boiling over means the water is coming out, you need to take the lid off and turn down the gas! Quickly! Boil over is also used as a metaphor when e.g. tension, or a hostile atmosphere gets so bad that the situation reaches the point of argument or even violence.

The two questions are very similar. The main difference being: The soup is boiling over: means that the water is coming over the edge of the pot. The soup is boiling up: means that the water is heating up but not yet boiling. The soup is boiling up is not a phrase you would hear very often. You would just say, the soup is boiling.

I've never heard the expression that anything is boiling up. I have heard... The soup is heating up. The soup is warming up. The soup is boiling over. I have some water on the boil (ready for the vegetables)