"Grow" and "grow up" can both mean maturing - children can grow or grow up to be adults. However, if you are talking about the purely physical process of growing - plants, for example - then you would just use "grow". "He grew up in the U.S." "He has grown since I last saw him - both in height and maturity." "These flowers grow well in this climate."

grow - everything grows grow up = become an adult; to mature