They have the same meaning. The only difference is who you are talking to.
The word 'grown-up' is mainly used when adults are talking to children, or when small children are talking among themselves. 'Grown-ups' is used in much the same way as French people say 'les grands' when they are talking to children about adults.
We also have the adjective 'grown-up' to mean the opposite of 'childish'. For example, if a child acts in a mature way, you could say that they are 'being very grown-up'. Adults can sometimes use this childhood language in a slightly jokey way. For example, when a twenty-something buys his or her first business suit to wear for work, they might describe it as 'grown-up' clothes.