they mean the same thing, just different ways of saying the same thing, but to grow rich implies that it happened very intentionally and over the course of time, and to become rich could have happened over time but implies it happened suddenly, I bought a lotto ticket and became rich over night, or my father died and I inherited a vast fortune and became rich. my father hired me in to the family business eventually turning it over to me after many years so I grew rich over the years.

To become rich makes you experience two different ways of living. You decide which of them is better for you, you and only you know what makes you happier and satisfied. Your old life provides you with experience, knowledge and strategies which will help you to succeed in your future. On the other hand, if you are born rich, everything is there for you. you only see one picture of the whole album. So if things don't go well and you are suddenly expelled out of it, you take a long time to adapt with the new surrounding circumstances. I am not saying that this is the rule but its just an opinion.

As far as I know the second one means to be rich after being a poor, however ,the first means you are rich from the start for instance to be born in a rich family. that is what i know ^^