"Grow up" is much more specific phrase than the verb "grow." Although "grow up" refers to the process of a child maturing into an adult, you can say to an adult, "Grow up!" In this context, it is used pejoratively, implying that the adult is behaving like a child. (Perhaps this command form is what Will had in mind when he wrote that this phrase is not used with a subject? You wouldn't say "Grow!" to a person...but I could imagine someone who likes to talk to their plants saying it.) Grow, as a more general word, can refer to lots of types of growth, including emotional, mental or spiritual growth. Also, as a more general word, it can imply a maturation process but not one that involves the change from childhood to adulthood. So here's two examples: "This experience has made me grow up." This sentences implies that I was immature--or childish--before (even if I was technically an adult), but now I act more like an adult. "The experience really helped me grow." This sentence can imply that I was a mature adult before, but now my outlook on life/emotional or spiritual awareness/perspectives, etc. have changed or expanded.

The rule of a thumb is that if there is a subject, you would use grow. If there is no subject you would use grow up. But there is exceptions to this rule. The way that I like to think about this is, the experience has made your mind grow. Mind is a subject thus you use grow.

Hi Simone. I would say, " This experience has made me grow. "

And to answer your other question. When we use this sentence it is related to mental growth, or maturity. This means that after you had this experience you have grown or matured from it.