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.