It's an interesting one I think.
Growing has so many ideas attached to it - plants, families, communities and so on.
Growing up has so many ideas attached to it - children, adults who should have but haven't, adults who have but choose to retain the best bits of childhood, maturity, and so on.
As per usual, here is the dictionary definition!
The American Heritage® Dictionary of the English Language, Fourth Edition copyright ©2000 by Houghton Mifflin Company. Updated in 2003. Published by Houghton Mifflin Company. All rights reserved.
Collins Essential English Dictionary 2nd Edition 2006 © HarperCollins Publishers 2004, 2006
Plenty to think about there I think!