Both mean mostly the same but are not very interchangeable.
A general distinction would be
* is casual, means literally something is running about. Sometimes it has negative connotation of "in vain".
* is often used for spreading of news. It can take animate subjects, but sounds a bit strained or affected. It usually means "purposefully going to many places".
**Examples** :
* The child ran about in a supermarket.
Using here is odd. It sounds like the child is doing so for a specific purpose.
* The news spread all over the world.
Using is much less idiomatic.
* I travelled all around the world.
This is more likely to be used than , which sounds like a sentence in an autobiography.
* I went everywhere (asking for money) but was unable to collect money.
Using is not possible.