Both phrases mean taking care of one's children, but I wonder whether there is any difference in nuance between them?
I disagree that both sentences can be translated as "taking care of the children". Note that in both cases you have to translate with the children and not one's children. There is no connotation in the German sentences regarding which children are acted with/upon.
The first sentence "sich mit den Kindern beschäftigen" has a more specific meaning, saying that you engage in some sort of activity with or for the children (beschäftigen -> occupy, engage). The second sentence "sich um die Kinder kümmern" has a wider sense and can be translated as "taking care of the children", but does not require an active part on your side.
For example, if you sit reading a book, having just an eye on your playing children, you do "dich um die Kinder kümmern", but you do not "dich mit den Kindern beschäftigen".