The difference in meaning is: "mon nouveau chapeau" where the hat is new to me but possibly a second hand one (but it can be new as well) and "mon chapeau neuf" if the hat has just been bought (no alternative).
hugolemieux I'm not agree. "mon nouveau chapeau" mean it's new for you, you just buy it, get it even if it was used. It's still a new stuff for you. "mon chapeau neuf" actually "mon chapeau est neuf" neam it's you're hat and it's new (never used)