"Grow up" está no presente.
Ex> She needs to grow up.
Stop acting like a child and grow up!
Já "grew up" é o passado desse verbo .
>> She grew up a lot when her father was fighting the war.
E 'grown up' é o particípio. é usado em tempos como present perfect,past perfect etc..
"They've grown up a lot since you last saw them"
"I saw them. They'd grown up and weren't kids anymore..."
E ainda, "grown-up" pode ser um adjetivo.
"He's a grown-up man"
Espero ter ajudado um pouco =)